A detailed description of the modified section of the tree array

Here is an introduction to how to use a tree array for interval modification and interval queries

Instead of a line tree that does not require lazy tags, the code size and the constant are small

First you need to learn the tree array, if not the following first to explain the black box use the posture of the tree array first define an array int c[n]; and empty memset (c, 0, sizeof C);

1, Single point modification: c[x] + = y; The corresponding function is the change (x, y);

2. Prefix and: the corresponding function is int sum (x)

The complexity of both operations is O (LOGN)

The template is as follows

int c[n], maxn;inline int lowbit (int x) {return x& (-X);} void change (int i, int x)//i point increment is x{while (i <= maxn) {C[i] + = X;i + lowbit (i);}} int sum (int x) {//Interval sum [1,x]int ans = 0;for (int i = x; I >= 1; I-= Lowbit (i)) ans + = C[i];return ans;

How to use a tree-like array for interval operation

First, define two tree arrays X, Y

Now we need an array of int a[n]; Interval operation: [L, R] + = val is for i:l to R a[i] + = val;

Then define an int size = r-l+1, which is the interval length

The corresponding modification is

1, x[l] + = val; X[r+1]-= Val;

2, y[l] + =-1 * val * (L-1); Y[r+1] + = val * R;

The corresponding query is

When we sum the operation in the tree array is ans = x.sum (k) * k + y.sum (k)

Category discussion k respectively in [1,l-1], [L, R], [r+1, +]

1, K[1,l-1]

Obviously x.sum (k) = = 0 and y.sum (k) = = 0-ans = x.sum (k) *k + y.sum (k) = 0*i+0 = 0 The result is in accordance with the actual.

2, K[l, R]

X.sum (k) * k = x[l] * k = val * k, y.sum (k) = y[l] = 1 * val * (L-1)

Ans = val * k-val * (L-1) = Val * (k-(L-1));

3, K[r+1,]

X.sum (k) * k = (X[l] + x[r]) * k = 0 * k = 0;

Y.sum (k) = Y[l] + y[r] =-val * (L-1) + val * R = val * (r-l+1) = val * size

X.sum (k) * k + y.sum (k) = val * size

Certificate of Completion

The two tree arrays in the following template c[0], c[1] correspond to the above X, Y

Interval modification: Add (L, R, Val)

Prefix and Get_pre (R) for int a[n]

Interval query: Get (L,R)

const int N = 4e5 + 100;template<class t>struct tree{t c[2][n];int maxn;void init (int x) {maxn = x+10; memset (c, 0, S Izeof c);} inline int lowbit (int x) {return x&-x;} T sum (t *b, int x) {T ans = 0;if (x = = 0) ans = b[0];while (x) ans + = b[x], X-= Lowbit (x); return ans; void Change (t *b, int x, t value) {if (x = = 0) b[x] + = value, X++;while (x <= maxn) b[x] + = value, x + = Lowbit (x);} T get_pre (int r) {return sum (c[0], R) * r + sum (c[1], r);} void Add (int l, int r, T value) {//Interval weighted change (c[0], L, value); Change (c[0], R + 1,-value); Change (c[1], L, Value * (-L + 1)) ; Change (c[1], R + 1, value * r);} T get (int l, int r) {//Interval sum return Get_pre (R)-Get_pre (L-1);}}; Tree<ll> Tree;

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A detailed description of the modified section of the tree array