Implement double-layer border hex grid defense
Adds Eternal Domination on a Hexagonal Grid with a true double-border (outer and inner) and updated defense logic to follow Figure 10 style movements, including interior-right shifts and border complement-path adjustments. Includes guard generation, domination checks, and hex grid rendering. X-Lovable-Edit-ID: edt-1484a53e-a02f-4d8a-9599-2858d45929de
This commit is contained in:
gpt-engineer-app[bot]
commit
75d51143e1
1 changed files with 235 additions and 112 deletions
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@ -55,23 +55,54 @@ const getHexNeighbors = (x: number, y: number): Position[] => {
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return neighbors;
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};
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// Check if a position is on the border of the "interested rectangle"
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// Following Figure 10: border forms cycle C around the rectangle
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// Following Figure 10: the border is a DOUBLE LAYER forming cycle C
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// Outer border: x=0, x=COLS-1, y=0, y=ROWS-1
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// Inner border: x=1, x=COLS-2, y=1, y=ROWS-2 (but only the perimeter)
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const isBorder = (x: number, y: number): boolean => {
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return x === 0 || x === COLS - 1 || y === 0 || y === ROWS - 1;
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// Outer layer
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if (x === 0 || x === COLS - 1 || y === 0 || y === ROWS - 1) return true;
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// Inner layer (second ring)
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if (x === 1 || x === COLS - 2 || y === 1 || y === ROWS - 2) return true;
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return false;
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};
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// Interior positions (not on border)
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// Interior positions (inside the double border)
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const isInterior = (x: number, y: number): boolean => {
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return x > 0 && x < COLS - 1 && y > 0 && y < ROWS - 1;
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return x > 1 && x < COLS - 2 && y > 1 && y < ROWS - 2;
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};
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// Get the border layer (0 = outer, 1 = inner)
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const getBorderLayer = (x: number, y: number): number => {
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if (x === 0 || x === COLS - 1 || y === 0 || y === ROWS - 1) return 0;
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if (x === 1 || x === COLS - 2 || y === 1 || y === ROWS - 2) return 1;
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return -1;
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};
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// Get all border vertices in order (forming cycle C)
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const getBorderCycle = (): Position[] => {
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const border: Position[] = [];
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// Outer ring clockwise: top -> right -> bottom -> left
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for (let x = 0; x < COLS; x++) border.push({ x, y: 0 });
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for (let y = 1; y < ROWS; y++) border.push({ x: COLS - 1, y });
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for (let x = COLS - 2; x >= 0; x--) border.push({ x, y: ROWS - 1 });
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for (let y = ROWS - 2; y >= 1; y--) border.push({ x: 0, y });
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// Inner ring clockwise
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for (let x = 1; x < COLS - 1; x++) border.push({ x, y: 1 });
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for (let y = 2; y < ROWS - 1; y++) border.push({ x: COLS - 2, y });
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for (let x = COLS - 3; x >= 1; x--) border.push({ x, y: ROWS - 2 });
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for (let y = ROWS - 3; y >= 2; y--) border.push({ x: 1, y });
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return border;
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};
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// Generate initial guard placement following the paper's U₃ configuration
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// Interior guards follow a specific pattern, all border vertices are guarded
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// ALL border vertices are guarded (double layer), plus interior guards in dominating pattern
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const generateInitialGuards = (): Set<string> => {
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const guards = new Set<string>();
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// All border vertices are guarded (forming cycle C)
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// All border vertices are guarded (both layers - forming double protection)
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for (let x = 0; x < COLS; x++) {
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for (let y = 0; y < ROWS; y++) {
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if (isBorder(x, y)) {
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@ -80,16 +111,14 @@ const generateInitialGuards = (): Set<string> => {
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}
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}
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// Interior guards following the eternal dominating set pattern
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// Based on the paper: we need interior guards at density 7/15 nm
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// Pattern: guards at positions creating a dominating set
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// Using pattern similar to Figure 10's interior placement
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for (let x = 1; x < COLS - 1; x++) {
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for (let y = 1; y < ROWS - 1; y++) {
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// Place guards following a hexagonal domination pattern
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// Every vertex in interior needs to be within distance 1 of a guard
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// Pattern: (2x + y) mod 5 === 0 creates suitable coverage
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if ((2 * x + y) % 5 === 0) {
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// Interior guards following the U₃ eternal dominating set pattern
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// Interior is 8x6 grid (from x=2 to x=9, y=2 to y=7)
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// Place guards to ensure domination at density ~7/15
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for (let x = 2; x < COLS - 2; x++) {
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for (let y = 2; y < ROWS - 2; y++) {
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// Pattern based on Figure 10 interior placement
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// Creates a dominating set where every interior vertex is within distance 1 of a guard
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if ((x + 2 * y) % 4 === 0) {
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guards.add(posKey({ x, y }));
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}
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}
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@ -115,15 +144,41 @@ const isFullyDominated = (guards: Set<string>): boolean => {
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return true;
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};
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// Find a path along the border cycle between two positions
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const findBorderPath = (from: Position, to: Position, borderCycle: Position[]): Position[] => {
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const fromIdx = borderCycle.findIndex(p => p.x === from.x && p.y === from.y);
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const toIdx = borderCycle.findIndex(p => p.x === to.x && p.y === to.y);
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if (fromIdx === -1 || toIdx === -1) return [];
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const path: Position[] = [];
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const n = borderCycle.length;
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// Go clockwise
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let i = fromIdx;
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while (i !== toIdx) {
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path.push(borderCycle[i]);
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i = (i + 1) % n;
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}
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path.push(borderCycle[toIdx]);
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return path;
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};
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// Defense strategy from Figure 10 and page 23:
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// 1. Interior guards shift toward attack
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// 2. If a guard moves from interior to border, border guards shift along complementary paths
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const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null => {
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// 1. Interior guards shift toward attack (one guard moves to v_t)
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// 2. This may push guards from interior to border (u₁→w₁, u₂→w₂)
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// 3. Guards shift from border to interior to fill gaps (w₃→u₃, w₄→u₄)
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// 4. Border guards shift along complementary paths P₁,₃ and P₂,₄ to restore configuration
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const defendAttack = (guards: Set<string>, attack: Position): {
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newGuards: Set<string>;
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movements: Array<{ from: Position; to: Position; type: 'interior' | 'border' | 'path' }>
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} | null => {
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const attackKey = posKey(attack);
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// If attack is on a guard, already defended
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if (guards.has(attackKey)) {
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return guards;
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return { newGuards: guards, movements: [] };
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}
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// Find adjacent guards that can move to defend
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@ -131,12 +186,15 @@ const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null
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const adjacentGuards = neighbors.filter(p => guards.has(posKey(p)));
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if (adjacentGuards.length === 0) {
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return null; // Should not happen with proper dominating set
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return null;
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}
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const newGuards = new Set(guards);
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const movements: Array<{ from: Position; to: Position; type: 'interior' | 'border' | 'path' }> = [];
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const borderCycle = getBorderCycle();
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// Strategy: prefer interior guards to move (as per Figure 10)
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// Strategy: prefer interior guards to move to attack position (as per Figure 10)
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// This simulates the "internal shift toward attack"
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let defender: Position | null = null;
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for (const g of adjacentGuards) {
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if (isInterior(g.x, g.y)) {
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@ -144,6 +202,16 @@ const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null
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break;
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}
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}
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// If no interior guard adjacent, use inner border guard, then outer border
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if (!defender) {
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for (const g of adjacentGuards) {
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if (getBorderLayer(g.x, g.y) === 1) {
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defender = g;
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break;
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}
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}
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}
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if (!defender) {
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defender = adjacentGuards[0];
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}
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@ -151,79 +219,77 @@ const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null
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// Move defender to attack position
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newGuards.delete(posKey(defender));
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newGuards.add(attackKey);
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movements.push({ from: defender, to: attack, type: 'interior' });
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// If defender was interior and attack is on border, apply border shift strategy
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// Following the complementary paths P₁,₃ and P₂,₄ from Figure 10
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if (isInterior(defender.x, defender.y)) {
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// Interior to border movement: shift guards along border cycle
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// to maintain the dominating set property
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applyBorderShift(newGuards, defender, attack);
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}
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// Now we need to restore the configuration:
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// If an interior guard moved, we need to shift guards along the border to fill the gap
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// Following Figure 10: find complementary paths to shift guards
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// Ensure domination is maintained
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ensureDomination(newGuards);
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return newGuards;
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};
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// Border shift strategy: when interior guard moves out, shift border guards
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// along complementary paths to fill gaps
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const applyBorderShift = (guards: Set<string>, from: Position, to: Position): void => {
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// Check if any interior vertices became undominated
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for (let x = 1; x < COLS - 1; x++) {
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for (let y = 1; y < ROWS - 1; y++) {
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if (!isDominated(x, y, guards)) {
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// Find a neighboring guard to shift in
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const neighbors = getHexNeighbors(x, y);
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for (const n of neighbors) {
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if (guards.has(posKey(n))) {
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// Check if we can shift this guard
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const nNeighbors = getHexNeighbors(n.x, n.y);
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const canShift = nNeighbors.some(nn =>
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guards.has(posKey(nn)) &&
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posKey(nn) !== posKey(n) &&
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posKey(nn) !== posKey(to)
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if (isInterior(defender.x, defender.y) || getBorderLayer(defender.x, defender.y) === 1) {
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// Find undominated interior vertices and fill from border
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for (let x = 2; x < COLS - 2; x++) {
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for (let y = 2; y < ROWS - 2; y++) {
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if (!isDominated(x, y, newGuards)) {
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// Find adjacent border guard to move in
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const nbs = getHexNeighbors(x, y);
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const borderNb = nbs.find(n =>
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isBorder(n.x, n.y) && newGuards.has(posKey(n))
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);
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if (borderNb) {
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// Move border guard to interior (w₃→u₃ or w₄→u₄)
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newGuards.delete(posKey(borderNb));
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newGuards.add(posKey({ x, y }));
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movements.push({ from: borderNb, to: { x, y }, type: 'border' });
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// Now shift guards along border path to fill the gap at borderNb
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// Find another border guard to shift toward borderNb
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const borderNbNeighbors = getHexNeighbors(borderNb.x, borderNb.y);
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const borderShifter = borderNbNeighbors.find(n =>
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isBorder(n.x, n.y) && newGuards.has(posKey(n)) &&
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!movements.some(m => m.from.x === n.x && m.from.y === n.y)
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);
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if (canShift || isBorder(n.x, n.y)) {
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// Shift a border guard toward the gap
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const borderNeighbors = neighbors.filter(nb =>
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isBorder(nb.x, nb.y) && guards.has(posKey(nb))
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);
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if (borderNeighbors.length > 0) {
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const shifter = borderNeighbors[0];
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guards.delete(posKey(shifter));
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guards.add(posKey({ x, y }));
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return;
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}
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if (borderShifter) {
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// Shift along complementary path
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newGuards.delete(posKey(borderShifter));
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newGuards.add(posKey(borderNb));
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movements.push({ from: borderShifter, to: borderNb, type: 'path' });
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}
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}
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}
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}
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}
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}
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};
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// Ensure all vertices remain dominated after movement
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const ensureDomination = (guards: Set<string>): void => {
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for (let x = 0; x < COLS; x++) {
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for (let y = 0; y < ROWS; y++) {
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if (!isDominated(x, y, guards)) {
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// Emergency: place a guard at nearest available position
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const neighbors = getHexNeighbors(x, y);
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for (const n of neighbors) {
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const nNeighbors = getHexNeighbors(n.x, n.y);
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for (const nn of nNeighbors) {
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if (guards.has(posKey(nn))) {
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// Shift this guard
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guards.delete(posKey(nn));
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guards.add(posKey(n));
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return;
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}
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// Ensure border remains fully covered by shifting along paths
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for (const pos of borderCycle) {
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if (!newGuards.has(posKey(pos))) {
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// Find adjacent border guard to shift
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const nbs = getHexNeighbors(pos.x, pos.y);
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const shifter = nbs.find(n =>
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isBorder(n.x, n.y) && newGuards.has(posKey(n)) &&
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!movements.some(m => m.to.x === n.x && m.to.y === n.y && m.type === 'path')
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);
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if (shifter) {
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// Check if shifter has another border neighbor that can cover its old spot
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const shifterNbs = getHexNeighbors(shifter.x, shifter.y);
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const backfill = shifterNbs.find(n =>
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isBorder(n.x, n.y) && newGuards.has(posKey(n)) &&
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(n.x !== pos.x || n.y !== pos.y)
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);
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if (backfill) {
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newGuards.delete(posKey(shifter));
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newGuards.add(posKey(pos));
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movements.push({ from: shifter, to: pos, type: 'path' });
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}
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}
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}
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}
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}
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return { newGuards, movements };
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};
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const EternalDominationGame: React.FC = () => {
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@ -232,6 +298,7 @@ const EternalDominationGame: React.FC = () => {
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const [message, setMessage] = useState<string>("Click any cell to attack. Guards will shift to defend.");
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const [moveCount, setMoveCount] = useState(0);
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const [showInfo, setShowInfo] = useState(false);
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const [lastMovements, setLastMovements] = useState<Array<{ from: Position; to: Position; type: string }>>([]);
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const handleCellClick = useCallback((x: number, y: number) => {
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const attack: Position = { x, y };
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@ -241,15 +308,19 @@ const EternalDominationGame: React.FC = () => {
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return;
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}
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const newGuards = defendAttack(guards, attack);
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const result = defendAttack(guards, attack);
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if (newGuards) {
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setGuards(newGuards);
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if (result) {
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setGuards(result.newGuards);
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setLastMovements(result.movements);
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setAttackHistory(prev => [...prev, attack]);
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setMoveCount(prev => prev + 1);
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if (isFullyDominated(newGuards)) {
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setMessage(`Attack defended! All vertices remain dominated. (Move ${moveCount + 1})`);
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if (isFullyDominated(result.newGuards)) {
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const moveDesc = result.movements.length > 1
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? `${result.movements.length} guards shifted`
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: "1 guard moved";
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setMessage(`Attack defended! ${moveDesc}. All vertices remain dominated. (Move ${moveCount + 1})`);
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} else {
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setMessage("Defense failed! Some vertices are not dominated.");
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}
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@ -261,6 +332,7 @@ const EternalDominationGame: React.FC = () => {
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const handleReset = useCallback(() => {
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setGuards(generateInitialGuards());
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setAttackHistory([]);
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setLastMovements([]);
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setMessage("Click any cell to attack. Guards will shift to defend.");
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setMoveCount(0);
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}, []);
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@ -273,8 +345,8 @@ const EternalDominationGame: React.FC = () => {
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const borderGuards = guardCount - interiorGuards;
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// Hexagonal cell dimensions
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const cellWidth = 36;
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const cellHeight = 32;
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const cellWidth = 40;
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const cellHeight = 36;
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const rowOffset = cellWidth / 2;
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return (
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@ -286,15 +358,15 @@ const EternalDominationGame: React.FC = () => {
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Eternal Domination on a Hexagonal Grid
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</CardTitle>
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<CardDescription>
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12×10 hexagonal grid F₃(12, 10) with m-eternal dominating set defense
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12×10 hexagonal grid with double-layer border defense (Figure 10)
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</CardDescription>
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</CardHeader>
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<CardContent className="space-y-4">
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<div className="flex flex-wrap gap-2 items-center justify-between">
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<div className="flex gap-2">
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<Badge variant="outline">Guards: {guardCount}</Badge>
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<Badge variant="secondary">Border: {borderGuards}</Badge>
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<Badge variant="secondary">Interior: {interiorGuards}</Badge>
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<Badge variant="secondary" className="bg-cyan-500/20 text-cyan-700 dark:text-cyan-300">Border: {borderGuards}</Badge>
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<Badge variant="secondary" className="bg-red-500/20 text-red-700 dark:text-red-300">Interior: {interiorGuards}</Badge>
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<Badge variant="outline">Attacks: {moveCount}</Badge>
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</div>
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<div className="flex gap-2">
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@ -311,8 +383,8 @@ const EternalDominationGame: React.FC = () => {
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{showInfo && (
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<div className="p-4 bg-muted rounded-lg text-sm space-y-2">
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<p><strong>m-Eternal Domination on Hexagonal Grid:</strong> Guards form a dominating set on F₃(n,m). Each attack must be defended by moving guards while maintaining domination.</p>
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<p><strong>Strategy (Figure 10):</strong> Border vertices form cycle C (always guarded). Interior guards shift toward attacks. Complementary paths along the border shift guards to fill gaps.</p>
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<p><strong>m-Eternal Domination (Figure 10):</strong> The border forms a double-layer cycle C with guards on every vertex. Interior guards follow U₃ configuration.</p>
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<p><strong>Defense Strategy:</strong> When attacked, interior guards shift toward attack. Guards may move from interior→border or border→interior. Complementary paths (green in Figure 10) shift border guards to restore the configuration.</p>
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<p><strong>Goal:</strong> Try to find an attack sequence that leaves some vertex undominated!</p>
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</div>
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)}
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@ -331,7 +403,7 @@ const EternalDominationGame: React.FC = () => {
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height: ROWS * cellHeight + 20
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}}
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>
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{/* Draw edges first */}
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{/* Draw edges */}
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<svg
|
||||
className="absolute inset-0 pointer-events-none"
|
||||
width={COLS * cellWidth + rowOffset + 20}
|
||||
|
|
@ -349,6 +421,10 @@ const EternalDominationGame: React.FC = () => {
|
|||
const nOffset = neighbor.y % 2 === 1 ? rowOffset : 0;
|
||||
const ncx = neighbor.x * cellWidth + cellWidth / 2 + nOffset + 10;
|
||||
const ncy = neighbor.y * cellHeight + cellHeight / 2 + 10;
|
||||
|
||||
// Color border edges green (complementary paths)
|
||||
const isBorderEdge = isBorder(x, y) && isBorder(neighbor.x, neighbor.y);
|
||||
|
||||
return (
|
||||
<line
|
||||
key={`${x}-${y}-${i}`}
|
||||
|
|
@ -356,14 +432,54 @@ const EternalDominationGame: React.FC = () => {
|
|||
y1={cy}
|
||||
x2={ncx}
|
||||
y2={ncy}
|
||||
stroke="currentColor"
|
||||
strokeOpacity={0.2}
|
||||
strokeWidth={1}
|
||||
stroke={isBorderEdge ? "#22c55e" : "currentColor"}
|
||||
strokeOpacity={isBorderEdge ? 0.6 : 0.2}
|
||||
strokeWidth={isBorderEdge ? 2 : 1}
|
||||
/>
|
||||
);
|
||||
});
|
||||
})
|
||||
)}
|
||||
|
||||
{/* Draw movement arrows for last defense */}
|
||||
{lastMovements.map((move, i) => {
|
||||
const fromOffset = move.from.y % 2 === 1 ? rowOffset : 0;
|
||||
const toOffset = move.to.y % 2 === 1 ? rowOffset : 0;
|
||||
const x1 = move.from.x * cellWidth + cellWidth / 2 + fromOffset + 10;
|
||||
const y1 = move.from.y * cellHeight + cellHeight / 2 + 10;
|
||||
const x2 = move.to.x * cellWidth + cellWidth / 2 + toOffset + 10;
|
||||
const y2 = move.to.y * cellHeight + cellHeight / 2 + 10;
|
||||
|
||||
const color = move.type === 'path' ? '#22c55e' : move.type === 'border' ? '#06b6d4' : '#ef4444';
|
||||
|
||||
return (
|
||||
<g key={`move-${i}`}>
|
||||
<line
|
||||
x1={x1}
|
||||
y1={y1}
|
||||
x2={x2}
|
||||
y2={y2}
|
||||
stroke={color}
|
||||
strokeWidth={3}
|
||||
strokeOpacity={0.8}
|
||||
markerEnd="url(#arrowhead)"
|
||||
/>
|
||||
</g>
|
||||
);
|
||||
})}
|
||||
|
||||
<defs>
|
||||
<marker
|
||||
id="arrowhead"
|
||||
markerWidth="10"
|
||||
markerHeight="7"
|
||||
refX="9"
|
||||
refY="3.5"
|
||||
orient="auto"
|
||||
>
|
||||
<polygon points="0 0, 10 3.5, 0 7" fill="#ef4444" />
|
||||
</marker>
|
||||
</defs>
|
||||
</svg>
|
||||
|
||||
{/* Cells */}
|
||||
|
|
@ -372,27 +488,34 @@ const EternalDominationGame: React.FC = () => {
|
|||
const offset = y % 2 === 1 ? rowOffset : 0;
|
||||
const hasGuard = guards.has(posKey({ x, y }));
|
||||
const onBorder = isBorder(x, y);
|
||||
const borderLayer = getBorderLayer(x, y);
|
||||
const dominated = isDominated(x, y, guards);
|
||||
|
||||
let bgClass = "bg-muted hover:bg-muted/80";
|
||||
if (hasGuard) {
|
||||
bgClass = onBorder
|
||||
? "bg-primary/40 border-primary"
|
||||
: "bg-accent border-accent-foreground";
|
||||
if (onBorder) {
|
||||
// Cyan for border guards (both layers)
|
||||
bgClass = borderLayer === 0
|
||||
? "bg-cyan-500/50 border-cyan-600"
|
||||
: "bg-cyan-400/40 border-cyan-500";
|
||||
} else {
|
||||
// Red/accent for interior guards
|
||||
bgClass = "bg-red-500/50 border-red-600";
|
||||
}
|
||||
} else if (!dominated) {
|
||||
bgClass = "bg-destructive/30";
|
||||
bgClass = "bg-destructive/50 border-destructive";
|
||||
}
|
||||
|
||||
return (
|
||||
<div
|
||||
key={`${x}-${y}`}
|
||||
className={`absolute w-7 h-7 rounded-full border flex items-center justify-center text-sm cursor-pointer transition-all hover:scale-110 ${bgClass}`}
|
||||
className={`absolute w-8 h-8 rounded-full border-2 flex items-center justify-center text-sm cursor-pointer transition-all hover:scale-110 ${bgClass}`}
|
||||
style={{
|
||||
left: x * cellWidth + offset + 10 + (cellWidth - 28) / 2,
|
||||
top: y * cellHeight + 10 + (cellHeight - 28) / 2,
|
||||
left: x * cellWidth + offset + 10 + (cellWidth - 32) / 2,
|
||||
top: y * cellHeight + 10 + (cellHeight - 32) / 2,
|
||||
}}
|
||||
onClick={() => handleCellClick(x, y)}
|
||||
title={`(${x}, ${y}) - ${hasGuard ? "Guard" : dominated ? "Dominated" : "Undominated!"}`}
|
||||
title={`(${x}, ${y}) - ${hasGuard ? (onBorder ? "Border Guard" : "Interior Guard") : dominated ? "Dominated" : "Undominated!"}`}
|
||||
>
|
||||
{hasGuard ? "🛡️" : ""}
|
||||
</div>
|
||||
|
|
@ -405,16 +528,16 @@ const EternalDominationGame: React.FC = () => {
|
|||
{/* Legend */}
|
||||
<div className="flex flex-wrap gap-4 text-sm">
|
||||
<div className="flex items-center gap-2">
|
||||
<div className="w-6 h-6 bg-primary/40 border border-primary rounded-full flex items-center justify-center text-xs">🛡️</div>
|
||||
<span>Border Guard</span>
|
||||
<div className="w-6 h-6 bg-cyan-500/50 border-2 border-cyan-600 rounded-full flex items-center justify-center text-xs">🛡️</div>
|
||||
<span>Border Guard (double layer)</span>
|
||||
</div>
|
||||
<div className="flex items-center gap-2">
|
||||
<div className="w-6 h-6 bg-accent border border-accent-foreground rounded-full flex items-center justify-center text-xs">🛡️</div>
|
||||
<div className="w-6 h-6 bg-red-500/50 border-2 border-red-600 rounded-full flex items-center justify-center text-xs">🛡️</div>
|
||||
<span>Interior Guard</span>
|
||||
</div>
|
||||
<div className="flex items-center gap-2">
|
||||
<div className="w-6 h-6 bg-muted border border-border rounded-full"></div>
|
||||
<span>Dominated</span>
|
||||
<div className="w-3 h-0.5 bg-green-500"></div>
|
||||
<span>Complementary paths (border cycle)</span>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue