10.2 Binary Search Insertion Point¶
Binary search can be used not only to search for target elements, but also to solve many variant problems, such as finding the insertion position of a target element.
10.2.1 Case Without Duplicate Elements¶
Question
Given a sorted array nums of length \(n\) and an element target, where the array contains no duplicate elements, insert target into nums while maintaining its sorted order. If target already exists in the array, insert it to its left. Return the index of target after insertion. An example is shown below.
Figure 10-4 Binary search insertion point example data
If we want to reuse the binary search code from the previous section, we need to answer the following two questions.
Question 1: When the array contains target, is the insertion point index the same as that element's index?
The problem requires inserting target to the left of equal elements, which means the newly inserted target replaces the position of the original target. In other words, when the array contains target, the insertion point index is the index of that target.
Question 2: When the array does not contain target, what is the insertion point index?
To analyze this further, consider the binary search process: when nums[m] < target, \(i\) moves, meaning that pointer \(i\) is approaching elements greater than or equal to target. Similarly, pointer \(j\) is always approaching elements less than or equal to target.
Therefore, when the binary search ends, \(i\) must point to the first element greater than target, and \(j\) must point to the first element less than target. It follows that when the array does not contain target, the insertion index is \(i\). The code is shown below:
binary_search_insertion.py
def binary_search_insertion_simple(nums: list[int], target: int) -> int:
"""Binary search for insertion point (no duplicate elements)"""
i, j = 0, len(nums) - 1 # Initialize closed interval [0, n-1]
while i <= j:
m = (i + j) // 2 # Calculate midpoint index m
if nums[m] < target:
i = m + 1 # target is in the interval [m+1, j]
elif nums[m] > target:
j = m - 1 # target is in the interval [i, m-1]
else:
return m # Found target, return insertion point m
# Target not found, return insertion point i
return i
binary_search_insertion.cpp
/* Binary search for insertion point (no duplicate elements) */
int binarySearchInsertionSimple(vector<int> &nums, int target) {
int i = 0, j = nums.size() - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) / 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
return m; // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i;
}
binary_search_insertion.java
/* Binary search for insertion point (no duplicate elements) */
int binarySearchInsertionSimple(int[] nums, int target) {
int i = 0, j = nums.length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) / 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
return m; // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i;
}
binary_search_insertion.cs
/* Binary search for insertion point (no duplicate elements) */
int BinarySearchInsertionSimple(int[] nums, int target) {
int i = 0, j = nums.Length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) / 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
return m; // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i;
}
binary_search_insertion.go
/* Binary search for insertion point (no duplicate elements) */
func binarySearchInsertionSimple(nums []int, target int) int {
// Initialize closed interval [0, n-1]
i, j := 0, len(nums)-1
for i <= j {
// Calculate the midpoint index m
m := i + (j-i)/2
if nums[m] < target {
// target is in the interval [m+1, j]
i = m + 1
} else if nums[m] > target {
// target is in the interval [i, m-1]
j = m - 1
} else {
// Found target, return insertion point m
return m
}
}
// Target not found, return insertion point i
return i
}
binary_search_insertion.swift
/* Binary search for insertion point (no duplicate elements) */
func binarySearchInsertionSimple(nums: [Int], target: Int) -> Int {
// Initialize closed interval [0, n-1]
var i = nums.startIndex
var j = nums.endIndex - 1
while i <= j {
let m = i + (j - i) / 2 // Calculate the midpoint index m
if nums[m] < target {
i = m + 1 // target is in the interval [m+1, j]
} else if nums[m] > target {
j = m - 1 // target is in the interval [i, m-1]
} else {
return m // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i
}
binary_search_insertion.js
/* Binary search for insertion point (no duplicate elements) */
function binarySearchInsertionSimple(nums, target) {
let i = 0,
j = nums.length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, use Math.floor() to round down
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
return m; // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i;
}
binary_search_insertion.ts
/* Binary search for insertion point (no duplicate elements) */
function binarySearchInsertionSimple(
nums: Array<number>,
target: number
): number {
let i = 0,
j = nums.length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, use Math.floor() to round down
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
return m; // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i;
}
binary_search_insertion.dart
/* Binary search for insertion point (no duplicate elements) */
int binarySearchInsertionSimple(List<int> nums, int target) {
int i = 0, j = nums.length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) ~/ 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
return m; // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i;
}
binary_search_insertion.rs
/* Binary search for insertion point (no duplicate elements) */
fn binary_search_insertion_simple(nums: &[i32], target: i32) -> i32 {
let (mut i, mut j) = (0, nums.len() as i32 - 1); // Initialize closed interval [0, n-1]
while i <= j {
let m = i + (j - i) / 2; // Calculate the midpoint index m
if nums[m as usize] < target {
i = m + 1; // target is in the interval [m+1, j]
} else if nums[m as usize] > target {
j = m - 1; // target is in the interval [i, m-1]
} else {
return m;
}
}
// Target not found, return insertion point i
i
}
binary_search_insertion.c
/* Binary search for insertion point (no duplicate elements) */
int binarySearchInsertionSimple(int *nums, int numSize, int target) {
int i = 0, j = numSize - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) / 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
return m; // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i;
}
binary_search_insertion.kt
/* Binary search for insertion point (no duplicate elements) */
fun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {
var i = 0
var j = nums.size - 1 // Initialize closed interval [0, n-1]
while (i <= j) {
val m = i + (j - i) / 2 // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1 // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1 // target is in the interval [i, m-1]
} else {
return m // Found target, return insertion point m
}
}
// Target not found, return insertion point i
return i
}
binary_search_insertion.rb
### Binary search insertion point (no duplicates) ###
def binary_search_insertion_simple(nums, target)
# Initialize closed interval [0, n-1]
i, j = 0, nums.length - 1
while i <= j
# Calculate the midpoint index m
m = (i + j) / 2
if nums[m] < target
i = m + 1 # target is in the interval [m+1, j]
elsif nums[m] > target
j = m - 1 # target is in the interval [i, m-1]
else
return m # Found target, return insertion point m
end
end
i # Target not found, return insertion point i
end
10.2.2 Case with Duplicate Elements¶
Question
Based on the previous problem, assume the array may contain duplicate elements, with everything else remaining the same.
Suppose there are multiple target elements in the array. Ordinary binary search can only return the index of one target, and cannot determine how many target elements are to the left and right of that element.
The problem requires inserting the target element at the leftmost position, so we need to find the index of the leftmost target in the array. A straightforward initial approach is to follow the steps shown in Figure 10-5:
- Perform binary search to obtain the index of any
target, denoted as \(k\). - Starting from index \(k\), perform linear traversal to the left, and return when the leftmost
targetis found.
Figure 10-5 Linear search for insertion point of duplicate elements
Although this method works, it includes linear search, resulting in a time complexity of \(O(n)\). When the array contains many duplicate target elements, this method is very inefficient.
Now consider extending the binary search code. As shown in Figure 10-6, the overall process remains unchanged: in each iteration, we first compute the midpoint index \(m\), then compare target with nums[m], leading to the following cases:
- When
nums[m] < targetornums[m] > target, it meanstargethas not been found yet, so use the standard interval-shrinking operation of binary search to move pointers \(i\) and \(j\) closer totarget. - When
nums[m] == target, it means elements less thantargetare in the interval \([i, m - 1]\), so use \(j = m - 1\) to shrink the interval, thereby moving pointer \(j\) closer to elements less thantarget.
After the loop completes, \(i\) points to the leftmost target, and \(j\) points to the first element less than target, so index \(i\) is the insertion point.
Figure 10-6 Steps for binary search insertion point of duplicate elements
Observe the following code: the branches nums[m] > target and nums[m] == target perform the same operation, so they can be merged.
Even so, we can still keep the conditional branches expanded, as the logic is clearer and more readable.
binary_search_insertion.py
def binary_search_insertion(nums: list[int], target: int) -> int:
"""Binary search for insertion point (with duplicate elements)"""
i, j = 0, len(nums) - 1 # Initialize closed interval [0, n-1]
while i <= j:
m = (i + j) // 2 # Calculate midpoint index m
if nums[m] < target:
i = m + 1 # target is in the interval [m+1, j]
elif nums[m] > target:
j = m - 1 # target is in the interval [i, m-1]
else:
j = m - 1 # The first element less than target is in the interval [i, m-1]
# Return insertion point i
return i
binary_search_insertion.cpp
/* Binary search for insertion point (with duplicate elements) */
int binarySearchInsertion(vector<int> &nums, int target) {
int i = 0, j = nums.size() - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) / 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
j = m - 1; // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i;
}
binary_search_insertion.java
/* Binary search for insertion point (with duplicate elements) */
int binarySearchInsertion(int[] nums, int target) {
int i = 0, j = nums.length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) / 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
j = m - 1; // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i;
}
binary_search_insertion.cs
/* Binary search for insertion point (with duplicate elements) */
int BinarySearchInsertion(int[] nums, int target) {
int i = 0, j = nums.Length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) / 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
j = m - 1; // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i;
}
binary_search_insertion.go
/* Binary search for insertion point (with duplicate elements) */
func binarySearchInsertion(nums []int, target int) int {
// Initialize closed interval [0, n-1]
i, j := 0, len(nums)-1
for i <= j {
// Calculate the midpoint index m
m := i + (j-i)/2
if nums[m] < target {
// target is in the interval [m+1, j]
i = m + 1
} else if nums[m] > target {
// target is in the interval [i, m-1]
j = m - 1
} else {
// The first element less than target is in the interval [i, m-1]
j = m - 1
}
}
// Return insertion point i
return i
}
binary_search_insertion.swift
/* Binary search for insertion point (with duplicate elements) */
func binarySearchInsertion(nums: [Int], target: Int) -> Int {
// Initialize closed interval [0, n-1]
var i = nums.startIndex
var j = nums.endIndex - 1
while i <= j {
let m = i + (j - i) / 2 // Calculate the midpoint index m
if nums[m] < target {
i = m + 1 // target is in the interval [m+1, j]
} else if nums[m] > target {
j = m - 1 // target is in the interval [i, m-1]
} else {
j = m - 1 // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i
}
binary_search_insertion.js
/* Binary search for insertion point (with duplicate elements) */
function binarySearchInsertion(nums, target) {
let i = 0,
j = nums.length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, use Math.floor() to round down
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
j = m - 1; // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i;
}
binary_search_insertion.ts
/* Binary search for insertion point (with duplicate elements) */
function binarySearchInsertion(nums: Array<number>, target: number): number {
let i = 0,
j = nums.length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, use Math.floor() to round down
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
j = m - 1; // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i;
}
binary_search_insertion.dart
/* Binary search for insertion point (with duplicate elements) */
int binarySearchInsertion(List<int> nums, int target) {
int i = 0, j = nums.length - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) ~/ 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
j = m - 1; // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i;
}
binary_search_insertion.rs
/* Binary search for insertion point (with duplicate elements) */
pub fn binary_search_insertion(nums: &[i32], target: i32) -> i32 {
let (mut i, mut j) = (0, nums.len() as i32 - 1); // Initialize closed interval [0, n-1]
while i <= j {
let m = i + (j - i) / 2; // Calculate the midpoint index m
if nums[m as usize] < target {
i = m + 1; // target is in the interval [m+1, j]
} else if nums[m as usize] > target {
j = m - 1; // target is in the interval [i, m-1]
} else {
j = m - 1; // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
i
}
binary_search_insertion.c
/* Binary search for insertion point (with duplicate elements) */
int binarySearchInsertion(int *nums, int numSize, int target) {
int i = 0, j = numSize - 1; // Initialize closed interval [0, n-1]
while (i <= j) {
int m = i + (j - i) / 2; // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1; // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1; // target is in the interval [i, m-1]
} else {
j = m - 1; // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i;
}
binary_search_insertion.kt
/* Binary search for insertion point (with duplicate elements) */
fun binarySearchInsertion(nums: IntArray, target: Int): Int {
var i = 0
var j = nums.size - 1 // Initialize closed interval [0, n-1]
while (i <= j) {
val m = i + (j - i) / 2 // Calculate the midpoint index m
if (nums[m] < target) {
i = m + 1 // target is in the interval [m+1, j]
} else if (nums[m] > target) {
j = m - 1 // target is in the interval [i, m-1]
} else {
j = m - 1 // The first element less than target is in the interval [i, m-1]
}
}
// Return insertion point i
return i
}
binary_search_insertion.rb
### Binary search insertion point (with duplicates) ###
def binary_search_insertion(nums, target)
# Initialize closed interval [0, n-1]
i, j = 0, nums.length - 1
while i <= j
# Calculate the midpoint index m
m = (i + j) / 2
if nums[m] < target
i = m + 1 # target is in the interval [m+1, j]
elsif nums[m] > target
j = m - 1 # target is in the interval [i, m-1]
else
j = m - 1 # The first element less than target is in the interval [i, m-1]
end
end
i # Return insertion point i
end
Tip
The code in this section uses the "closed interval" approach throughout. Interested readers can implement the "left-closed, right-open" approach themselves.
Overall, binary search is simply a matter of setting separate search targets for pointers \(i\) and \(j\). The target may be a specific element (such as target) or a range of elements (such as elements less than target).
With each iteration of binary search, pointers \(i\) and \(j\) gradually approach their preset targets. Ultimately, they either find the answer or stop after crossing the boundary.