16 Jul 2024•TypeScript•Hard

Stamping the Grid

check if a grid can be fully covered by stamps using prefix sums and 2d difference array.

first, build a prefix sum to quickly check if a stamp can be placed (area sum must be 0, meaning no obstacles). then use a 2d difference array to track which cells are covered by stamps.

after marking all possible stamp placements, accumulate the difference array to get actual coverage. every empty cell must be covered by at least one stamp.

steps

build prefix sum for O(1) area queries
mark all valid stamp positions in difference array
accumulate difference array to get coverage
verify all empty cells are covered

complexity

O(mn) time for building prefix, checking stamps, and verifying coverage. O(mn) space for the arrays. efficient grid processing.

Solution files

TypeScript•stamping-the-grid/solution.ts

function possibleToStamp(grid: number[][], stampHeight: number, stampWidth: number): boolean {
  const rows = grid.length;
  if (rows === class="syntax-number">0) {
    return true;
  }
  const cols = grid[class="syntax-number">0].length;

  const prefix = buildPrefix(grid);

  const canStamp = Array.from({ length: rows }, () => new Array<boolean>(cols).fill(false));

  for (let r = class="syntax-number">0; r + stampHeight - class="syntax-number">1 < rows; ++r) {
    for (let c = class="syntax-number">0; c + stampWidth - class="syntax-number">1 < cols; ++c) {
      const r2 = r + stampHeight - class="syntax-number">1;
      const c2 = c + stampWidth - class="syntax-number">1;

      if (areaSum(prefix, r, c, r2, c2) === class="syntax-number">0) {
        canStamp[r][c] = true;
      }
    }
  }

  const cover = Array.from({ length: rows + class="syntax-number">1 }, () => new Array<number>(cols + class="syntax-number">1).fill(class="syntax-number">0));

  for (let r = class="syntax-number">0; r < rows; ++r) {
    for (let c = class="syntax-number">0; c < cols; ++c) {
      if (!canStamp[r][c]) {
        continue;
      }

      cover[r][c] += class="syntax-number">1;
      cover[r + stampHeight][c] -= class="syntax-number">1;
      cover[r][c + stampWidth] -= class="syntax-number">1;
      cover[r + stampHeight][c + stampWidth] += class="syntax-number">1;
    }
  }

  for (let r = class="syntax-number">0; r < rows; ++r) {
    for (let c = class="syntax-number">0; c < cols; ++c) {
      cover[r][c + class="syntax-number">1] += cover[r][c];
    }
  }

  for (let c = class="syntax-number">0; c < cols; ++c) {
    for (let r = class="syntax-number">0; r < rows; ++r) {
      cover[r + class="syntax-number">1][c] += cover[r][c];
    }
  }

  for (let r = class="syntax-number">0; r < rows; ++r) {
    for (let c = class="syntax-number">0; c < cols; ++c) {
      if (grid[r][c] === class="syntax-number">1) {
        continue;
      }
      if (cover[r][c] <= class="syntax-number">0) {
        return false;
      }
    }
  }

  return true;
}

function buildPrefix(grid: number[][]): number[][] {
  const rows = grid.length;
  const cols = rows ? grid[class="syntax-number">0].length : class="syntax-number">0;
  const prefix = Array.from({ length: rows + class="syntax-number">1 }, () => new Array<number>(cols + class="syntax-number">1).fill(class="syntax-number">0));

  for (let r = class="syntax-number">0; r < rows; ++r) {
    for (let c = class="syntax-number">0; c < cols; ++c) {
      prefix[r + class="syntax-number">1][c + class="syntax-number">1] =
        grid[r][c] + prefix[r][c + class="syntax-number">1] + prefix[r + class="syntax-number">1][c] - prefix[r][c];
    }
  }

  return prefix;
}

function areaSum(prefix: number[][], r1: number, c1: number, r2: number, c2: number): number {
  return (
    prefix[r2 + class="syntax-number">1][c2 + class="syntax-number">1] -
    prefix[r1][c2 + class="syntax-number">1] -
    prefix[r2 + class="syntax-number">1][c1] +
    prefix[r1][c1]
  );
}

export { possibleToStamp };