Binary Search
二分查找 binary search
二分查找是一种基于分治策略的高效搜索算法。它利用数据的有序性，每轮缩小一半搜索范围，直至找到目标元素或搜索区间为空为止。
- Pros: 时间效率高，无需额外空间开销。
- Cons: 仅适用于有序数组
JAVA
/* 二分查找（双闭区间） */int binarySearch(int[] nums, int target) { // 初始化双闭区间 [0, n-1] ，即 i, j 分别指向数组首元素、尾元素 int i = 0, j = nums.length - 1; // 循环，当搜索区间为空时跳出（当 i > j 时为空） while (i <= j) { int m = i + (j - i) / 2; // 计算中点索引 m if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j] 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m-1] 中 j = m - 1; else // 找到目标元素，返回其索引 return m; } // 未找到目标元素，返回 -1 return -1;}/* 二分查找（左闭右开区间） */int binarySearchLCRO(int[] nums, int target) { // 初始化左闭右开区间 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1 int i = 0, j = nums.length; // 循环，当搜索区间为空时跳出（当 i = j 时为空） while (i < j) { int m = i + (j - i) / 2; // 计算中点索引 m if (nums[m] < target) // 此情况说明 target 在区间 [m+1, j) 中 i = m + 1; else if (nums[m] > target) // 此情况说明 target 在区间 [i, m) 中 j = m; else // 找到目标元素，返回其索引 return m; } // 未找到目标元素，返回 -1 return -1;}JAVA
//寻找左侧边界的二分搜索int left_bound(int[] nums, int target) { int left = 0; // 注意 int right = nums.length; // 注意 while (left < right) { int mid = left + (right - left) / 2; if (nums[mid] == target) { right = mid; } else if (nums[mid] < target) { left = mid + 1; } else if (nums[mid] > target) { // 注意 right = mid; } } return left;}//寻找右侧边界的二分搜索int right_bound(int[] nums, int target) { int left = 0, right = nums.length; while (left < right) { int mid = left + (right - left) / 2; if (nums[mid] == target) { left = mid + 1; } else if (nums[mid] < target) { left = mid + 1; } else if (nums[mid] > target) { right = mid; } } // 注意 return left - 1;}Examples
34-在排序数组中查找元素的第一个和最后一个位置
JAVA
class Solution { public int[] searchRange(int[] nums, int target) { int leftIdx = binarySearch(nums, target, true); int rightIdx = binarySearch(nums, target, false); if (leftIdx <= rightIdx && rightIdx < nums.length && nums[leftIdx] == target && nums[rightIdx] == target) { return new int[] { leftIdx, rightIdx }; } return new int[] { -1, -1 }; } public int binarySearch(int[] nums, int target, boolean lower) { int left = 0, right = nums.length; while (left < right) { int mid = left + (right - left) / 2; if (nums[mid] < target) { left = mid + 1; } else if (nums[mid] > target) { right = mid; } else if (nums[mid] == target) { if (lower) { right = mid; } else { left = mid + 1; } } } return lower ? left : left - 1; }}704-二分查找
Time Complexity $O(log_n)$
Space Complexity $O(1)$
JAVA
class Solution { public int search(int[] nums, int target) { int left = 0, right = nums.length; while (left < right) { int mid = left + (right - left) / 2; if (nums[mid] == target) { return mid; } else if (nums[mid] < target) { left = mid + 1; } else { right = mid - 1; } } return -1; }}875-爱吃香蕉的珂珂
JAVA
class Solution { public int minEatingSpeed(int[] piles, int H) { int left = 1, right = 1; for (int pile : piles) { right = Math.max(right, pile); } while (left < right) { int mid = left + (right - left) / 2; if (estimatedHours(piles, mid) > H) { left = mid + 1; } else { right = mid; } } return left; } private int estimatedHours(int[] piles, int speed) { int hours = 0; for (int pile : piles) { hours += pile % speed == 0 ? pile / speed : pile / speed + 1; } return hours; }}1011-在 D 天内送达包裹的能力
1011,875两题区别
1011：对任意 i，有 capacity > weight[i]，即一天至少运一箱货
875：对任意 i，有 pile[i] >= speed，即一小时最多吃一堆香蕉
JAVA
class Solution { public int shipWithinDays(int[] weights, int days) { int max = 0, sum = 0; for (int weight : weights) { max = Math.max(max, weight); sum += weight; } int left = max, right = sum; while (left < right) { int mid = left + (right - left) / 2; if (canDeliverWithinDays(weights, mid, days)) { // 对应mid[i] >= target right = mid; } else { left = mid + 1; } } return left; } boolean canDeliverWithinDays(int[] weight, int capacity, int days) { int currentWeightSum = 0, cntOfDay = 1; for (int w : weight) { currentWeightSum += w; if (currentWeightSum > capacity) { currentWeightSum = w; cntOfDay++; } } return cntOfDay <= days; }}Ref
Binary Search