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Author: pushpit-J19

Data Structures/Trees/segment_tree_range_sum_and_update_query.py

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+"""
+Contains program from Segment tree creation from an array, range sum query and update query for the same.
+
+
+Segment trees are special kind of trees that allows answering range queries and updating values in an array effectively (both in O(logn)), 
+For an array, its segment tree will have as leaf nodes = the elements of the array,
+and its internal nodes store the value of a preprocessed value, like the sum of all its children
+
+eg arr : [1,2,3,4]  
+parent node of (1,2) = 3, parent node of (3,4) = 7
+parent node of (3,7) = 10 (this will also be the root node)
+Equivalent segment tree : [0, 10, 3, 7, 1, 2, 3, 4]
+where 0 has been added as first element to ease the calculations
+
+
+Sum range query is the sum of all elements in an array over a given range (l, r) both included, where l and r are the indicies.
+eg for arr = [1,2,3,4], sum_range(0,2) = 1+2+3 = 6
+The above approach has time complexity : O(n) 
+It can be improved by using the segment trees.
+
+For sum range query, the internal nodes store the sum of their children nodes.
+With segment trees, instead of going to every element, we already have the sum calculated in the parent node, so we just need to get that.
+to get above example of sum_range(0,2) = 1+2+3 :-
+tree : [0, 10, 3, 7, 1, 2, 3, 4]
+index: [0,  1, 2, 3, 4, 5, 6, 7]
+l = 4, r = 6 are the tree indices
+elem 1 and 2 have a common parent, where elem 3 has no common parent with others, so we add tree[r] = 3 to range_sum and shift r left, 
+that is, we move r to left if it is even. Similarly, l will have to be moved forward if it is odd.
+now l = 4, r = 5, range_sum=3, and we move to parent of both l and r
+l = 2, r = 2, now r is even, so tree[r]=tree[2]=3 is added to range_sum. range_sum = 3+3 = 6, r = 1
+so we stop, range_sum is our answer
+
+
+On updating, say changing 1 -> -1 on index 0
+arr becomes = [-1, 2, 3, 4]
+
+we need to only update the parent nodes (which are at index/2 of a node). So, no of queries = height of tree O(log(n)), 
+instead of updating boxes in sqrt decomposition O(sqrt(n)), and all the sums in dp approach O(n)
+so updated tree = [0, 8, 1, 7, -1, 2, 3, 4]
+
+"""
+
+from math import ceil, log2
+
+
+def segment_tree_creation (arr):
+    """The function creates a segment tree in O(n) time
+        arguments : array, whose corresponding segment tree is to be made
+        returns : the correspoding segment tree in a list format
+    """
+    n = len(arr)
+    
+    # for n leaf nodes, there are maximum double of 2**max_height-1 nodes, adding extra 0th elem for ease of calculations 
+    height = ceil(log2(n))
+    m = 2*(2**height)
+    tree = [0]*m        
+    
+    # filling the second half of the tree list (leaf nodes) with the array elements
+    for i in range(n):
+        index = i+int(m/2)
+        tree[index] = arr[i]
+    
+    # creating the internal nodes, calculating the from leaf to the root
+    for i in range(int(m/2)-1,0,-1):
+        tree[i] = tree[2*i] + tree[2*i+1]
+    return tree
+
+
+def segment_tree_range_sum_query (tree, l, r):
+    """The function tells the sum of elements in a given range using the segment tree in O(log(n))
+        input : segment tree, array starting and ending index of range
+        returns the sum of elements of array in that range"""
+
+    m = len(tree)
+    l,r = l+m//2, r+m//2    # converting array index to segment tree index
+    rsum = 0
+    while (l<=r) :          # while the lower and upper limit do not cross each other
+        if l%2 == 1:        # if lower range index is a right child, add that and jump to next index (towards right)
+            rsum += tree[l]
+            l += 1
+        if r%2 == 0:        # if upper range index is a left child, add that and jump to next index (towards left)
+            rsum += tree[r]
+            r -= 1
+
+        l, r = l//2, r//2       # both the indices are have a common ancestor in tree, so move up, till that ancestor
+        
+    return rsum
+
+
+def segment_tree_update_query (arr, tree, index, value):
+    """The function updates the node in array (leaf node) and all its parent nodes till root node, complexity = O(log(n))
+        input: array to be updated, tree to be updated, index of array, new value
+        returns : updated array and the tree
+    """
+    arr[index] = value
+    m = len(tree)
+    i = m//2 + index        # getting the index according to the tree from array index
+    change = value - tree[i]
+    while i>0 :
+        tree[i] += change   # updating the current nodes
+        i = i//2        # going to update the parent 
+    return arr, tree
+
+
+# DRIVER CODE
+if __name__ == '__main__':
+    
+    arr = [5, 8, 6, 3, 2, 7, 4, 6]
+    print("\nOriginal array : ", arr)
+
+    seg_tree = segment_tree_creation(arr)
+    print("\nSegment tree : ", seg_tree)
+
+    range_sum = segment_tree_range_sum_query(seg_tree, 1, 7)    # 8 + 6 + 3 + 2 + 7 + 4 + 6 = 36
+    print("\nRange sum index 1-7 = ", range_sum)
+
+    arr, seg_tree = segment_tree_update_query(arr, seg_tree, 2, -5)       # changing to 6 -> -5 at index 2
+    print("\nUpdated array tree : ", arr)
+    print("Updated Segment tree : ", seg_tree)
+
+    range_sum = segment_tree_range_sum_query(seg_tree, 1, 7)    # 8 + (-5) + 3 + 2 + 7 + 4 + 6 = 25
+    print("\nRange sum index 1-7 = ", range_sum, "\n")