"content":"<p>Fruits are available at some positions on an infinite x-axis. You are given a 2D integer array <code>fruits</code> where <code>fruits[i] = [position<sub>i</sub>, amount<sub>i</sub>]</code> depicts <code>amount<sub>i</sub></code> fruits at the position <code>position<sub>i</sub></code>. <code>fruits</code> is already <strong>sorted</strong> by <code>position<sub>i</sub></code> in <strong>ascending order</strong>, and each <code>position<sub>i</sub></code> is <strong>unique</strong>.</p>\n\n<p>You are also given an integer <code>startPos</code> and an integer <code>k</code>. Initially, you are at the position <code>startPos</code>. From any position, you can either walk to the <strong>left or right</strong>. It takes <strong>one step</strong> to move <strong>one unit</strong> on the x-axis, and you can walk <strong>at most</strong> <code>k</code> steps in total. For every position you reach, you harvest all the fruits at that position, and the fruits will disappear from that position.</p>\n\n<p>Return <em>the <strong>maximum total number</strong> of fruits you can harvest</em>.</p>\n\n<p> </p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2021/11/21/1.png\" style=\"width: 472px; height: 115px;\" />\n<pre>\n<strong>Input:</strong> fruits = [[2,8],[6,3],[8,6]], startPos = 5, k = 4\n<strong>Output:</strong> 9\n<strong>Explanation:</strong> \nThe optimal way is to:\n- Move right to position 6 and harvest 3 fruits\n- Move right to position 8 and harvest 6 fruits\nYou moved 3 steps and harvested 3 + 6 = 9 fruits in total.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2021/11/21/2.png\" style=\"width: 512px; height: 129px;\" />\n<pre>\n<strong>Input:</strong> fruits = [[0,9],[4,1],[5,7],[6,2],[7,4],[10,9]], startPos = 5, k = 4\n<strong>Output:</strong> 14\n<strong>Explanation:</strong> \nYou can move at most k = 4 steps, so you cannot reach position 0 nor 10.\nThe optimal way is to:\n- Harvest the 7 fruits at the starting position 5\n- Move left to position 4 and harvest 1 fruit\n- Move right to position 6 and harvest 2 fruits\n- Move right to position 7 and harvest 4 fruits\nYou moved 1 + 3 = 4 steps and harvested 7 + 1 + 2 + 4 = 14 fruits in total.\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2021/11/21/3.png\" style=\"width: 476px; height: 100px;\" />\n<pre>\n<strong>Input:</strong> fruits = [[0,3],[6,4],[8,5]], startPos = 3, k = 2\n<strong>Output:</strong> 0\n<strong>Explanation:</strong>\nYou can move at most k = 2 steps and cannot reach any position with fruits.\n</pre>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 <= fruits.length <= 10<sup>5</sup></code></li>\n\t<li><code>fruits[i].length == 2</code></li>\n\t<li><code>0 <= startPos, position<sub>i</sub> <= 2 * 10<sup>5</sup></code></li>\n\t<li><code>position<sub>i-1</sub> < position<sub>i</sub></code> for any <code>i > 0</code> (<strong>0-indexed</strong>)</li>\n\t<li><code>1 <= amount<sub>i</sub> <= 10<sup>4</sup></code></li>\n\t<li><code>0 <= k <= 2 * 10<sup>5</sup></code></li>\n</ul>\n",
"Does an optimal path have very few patterns? For example, could a path that goes left, turns and goes right, then turns again and goes left be any better than a path that simply goes left, turns, and goes right?",
"The optimal path turns at most once. That is, the optimal path is one of these: to go left only; to go right only; to go left, turn and go right; or to go right, turn and go left.",
"Moving x steps left then k-x steps right gives you a range of positions that you can reach.",
"Use prefix sums to get the sum of all fruits for each possible range.",
"Use a similar strategy for all the paths that go right, then turn and go left."
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