Add top-down dynamic programming solution to Jump Game.

Author: trekhleb

src/algorithms/uncategorized/jump-game/__test__/dpTopDownJumpGame.test.js

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+import dpTopDownJumpGame from '../dpTopDownJumpGame';
+
+describe('dpTopDownJumpGame', () => {
+  it('should solve Jump Game problem in top-down dynamic programming manner', () => {
+    expect(dpTopDownJumpGame([1, 0])).toBeTruthy();
+    expect(dpTopDownJumpGame([100, 0])).toBeTruthy();
+    expect(dpTopDownJumpGame([2, 3, 1, 1, 4])).toBeTruthy();
+    expect(dpTopDownJumpGame([1, 1, 1, 1, 1])).toBeTruthy();
+    expect(dpTopDownJumpGame([1, 1, 1, 10, 1])).toBeTruthy();
+    expect(dpTopDownJumpGame([1, 5, 2, 1, 0, 2, 0])).toBeTruthy();
+
+    expect(dpTopDownJumpGame([1, 0, 1])).toBeFalsy();
+    expect(dpTopDownJumpGame([3, 2, 1, 0, 4])).toBeFalsy();
+    expect(dpTopDownJumpGame([0, 0, 0, 0, 0])).toBeFalsy();
+    expect(dpTopDownJumpGame([5, 4, 3, 2, 1, 0, 0])).toBeFalsy();
+  });
+});

src/algorithms/uncategorized/jump-game/dpTopDownJumpGame.js

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+/**
+ * DYNAMIC PROGRAMMING TOP-DOWN approach of solving Jump Game.
+ *
+ * This comes out as an optimisation of BACKTRACKING approach.
+ * Optimisation is done by using memo table where we store information
+ * about each cell whether it is "good" or "bad" or "unknown".
+ *
+ * We call a position in the array a "good" one if starting at that
+ * position, we can reach the last index.
+ *
+ * @param {number[]} numbers - array of possible jump length.
+ * @param {number} startIndex - index from where we start jumping.
+ * @param {number[]} currentJumps - current jumps path.
+ * @param {boolean[]} cellsGoodness - holds information about whether cell is "good" or "bad"
+ * @return {boolean}
+ */
+export default function dpTopDownJumpGame(
+  numbers,
+  startIndex = 0,
+  currentJumps = [],
+  cellsGoodness = [],
+) {
+  if (startIndex === numbers.length - 1) {
+    // We've jumped directly to last cell. This situation is a solution.
+    return true;
+  }
+
+  // Init cell goodness table if it is empty.
+  // This is DYNAMIC PROGRAMMING feature.
+  const currentCellsGoodness = [...cellsGoodness];
+  if (!currentCellsGoodness.length) {
+    numbers.forEach(() => currentCellsGoodness.push(undefined));
+    // Mark the last cell as "good" one since it is where
+    // we ultimately want to get.
+    currentCellsGoodness[cellsGoodness.length - 1] = true;
+  }
+
+  // Check what the longest jump we could make from current position.
+  // We don't need to jump beyond the array.
+  const maxJumpLength = Math.min(
+    numbers[startIndex], // Jump is within array.
+    numbers.length - 1 - startIndex, // Jump goes beyond array.
+  );
+
+  // Let's start jumping from startIndex and see whether any
+  // jump is successful and has reached the end of the array.
+  for (let jumpLength = maxJumpLength; jumpLength > 0; jumpLength -= 1) {
+    // Try next jump.
+    const nextIndex = startIndex + jumpLength;
+
+    // Jump only into "good" or "unknown" cells.
+    // This is top-down dynamic programming optimisation of backtracking algorithm.
+    if (currentCellsGoodness[nextIndex] !== false) {
+      currentJumps.push(nextIndex);
+
+      const isJumpSuccessful = dpTopDownJumpGame(
+        numbers,
+        nextIndex,
+        currentJumps,
+        currentCellsGoodness,
+      );
+
+      // Check if current jump was successful.
+      if (isJumpSuccessful) {
+        return true;
+      }
+
+      // BACKTRACKING.
+      // If previous jump wasn't successful then retreat and try the next one.
+      currentJumps.pop();
+
+      // Mark current cell as "bad" to avoid its deep visiting later.
+      currentCellsGoodness[nextIndex] = false;
+    }
+  }
+
+  return false;
+}