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Posts published in “Data Structure”

花花酱 307. Range Sum Query – Mutable



Given an integer array nums, find the sum of the elements between indices i and j (i ≤ j), inclusive.

The update(i, val) function modifies nums by updating the element at index i to val.



  1. The array is only modifiable by the update function.
  2. You may assume the number of calls to update and sumRange function is distributed evenly.


Fenwick Tree



Time complexity:

init O(nlogn)

query: O(logn)

update: O(logn)




Solution 2: Segment Tree


花花酱 LeetCode Disjoint set / Union Find Forest SP1

Disjoint-set/Union-find Forest

Find(x): find the root/cluster-id of x

Union(x, y): merge two clusters

Check whether two elements are in the same set or not in O(1)*.

Find: O(ɑ(n))* ≈ O(1)

Union: O(ɑ(n))* ≈ O(1)

Space: O(n)

Without optimization: Find: O(n)

Two key optimizations:

  1. Path compression: make tree flat
  2. Union by rank: merge low rank tree to high rank one

*: amortized

ɑ(.): inverse Ackermann function






Union-Find Problems


花花酱 LeetCode 715. Range Module


A Range Module is a module that tracks ranges of numbers. Your task is to design and implement the following interfaces in an efficient manner.


  • addRange(int left, int right) Adds the half-open interval [left, right), tracking every real number in that interval. Adding an interval that partially overlaps with currently tracked numbers should add any numbers in the interval [left, right) that are not already tracked.
  • queryRange(int left, int right) Returns true if and only if every real number in the interval [left, right) is currently being tracked.
  • removeRange(int left, int right) Stops tracking every real number currently being tracked in the interval [left, right).

Example 1:


  • A half open interval [left, right) denotes all real numbers left <= x < right.
  • 0 < left < right < 10^9 in all calls to addRange, queryRange, removeRange.
  • The total number of calls to addRange in a single test case is at most 1000.
  • The total number of calls to queryRange in a single test case is at most 5000.
  • The total number of calls to removeRange in a single test case is at most 1000.


map / ordered ranges




C++ / vector

C++ / map

Related Problems:

花花酱 LeetCode 208. Implement Trie (Prefix Tree)


Implement a trie with insertsearch, and startsWith methods.

You may assume that all inputs are consist of lowercase letters a-z.


Tree/children array




C++ / Array


C++ / hashmap




Python 1:


Python 2:


花花酱 LeetCode 218. The Skyline Problem


A city’s skyline is the outer contour of the silhouette formed by all the buildings in that city when viewed from a distance. Now suppose you are given the locations and height of all the buildings as shown on a cityscape photo (Figure A), write a program to output the skyline formed by these buildings collectively (Figure B).

Buildings Skyline Contour

The geometric information of each building is represented by a triplet of integers [Li, Ri, Hi], where Li and Ri are the x coordinates of the left and right edge of the ith building, respectively, and Hi is its height. It is guaranteed that 0 ≤ Li, Ri ≤ INT_MAX0 < Hi ≤ INT_MAX, and Ri - Li > 0. You may assume all buildings are perfect rectangles grounded on an absolutely flat surface at height 0.

For instance, the dimensions of all buildings in Figure A are recorded as: [ [2 9 10], [3 7 15], [5 12 12], [15 20 10], [19 24 8] ] .

The output is a list of “key points” (red dots in Figure B) in the format of [ [x1,y1], [x2, y2], [x3, y3], ... ] that uniquely defines a skyline. A key point is the left endpoint of a horizontal line segment. Note that the last key point, where the rightmost building ends, is merely used to mark the termination of the skyline, and always has zero height. Also, the ground in between any two adjacent buildings should be considered part of the skyline contour.

For instance, the skyline in Figure B should be represented as:[ [2 10], [3 15], [7 12], [12 0], [15 10], [20 8], [24, 0] ].


  • The number of buildings in any input list is guaranteed to be in the range [0, 10000].
  • The input list is already sorted in ascending order by the left x position Li.
  • The output list must be sorted by the x position.
  • There must be no consecutive horizontal lines of equal height in the output skyline. For instance, [...[2 3], [4 5], [7 5], [11 5], [12 7]...] is not acceptable; the three lines of height 5 should be merged into one in the final output as such: [...[2 3], [4 5], [12 7], ...]



Sweep line

Time Complexity:


Space Complexity:


Solution1: Heap 



Solution 2: Multiset