線段樹

線段樹

對於有一類的問題，我們主要關心的是線段(區間)，比如說查詢一個區間[i, j]內的最大值，最小值等等。假設你有一個網站，你想查詢某年(或某年以後)的用戶訪問量，消費最多的用戶等等，這些都是在某個區間內進行查詢，一般線段樹的區間是固定的，不包含刪除和添加的操作，只有查詢和更新的操作

線段樹的表示

現在如果假設有n個元素，用數組存儲的話，需要多少空間呢

public class SegmentTree<E> {
    private E[] tree;
    private E[] data;

    public SegmentTree(E[] arr) {
        data = (E[]) new Object[arr.length];
        for (int i = 0; i < arr.length; i++) {
            data[i] = arr[i];
        }

        tree = (E[]) new Object[4 * data.length];
    }

    public int getSize() {
        return data.length;
    }
    public E get(int index) {
        if (index < 0 || index >= data.length) {
            throw new IllegalArgumentException("參數錯誤");
        }
        return data[index];
    }
    
    private int leftChild(int index) {
        return 2 * index + 1;
    }
    private int rightChild(int index) {
        return 2 * index + 2;
    }
}

實現

創建線段樹

下面就要根據數組來創建一棵線段樹，我們的方法先創建下面的子線段樹，然後由這些子線段樹合併成大的線段樹，以此類推

在合併左右子樹的過程中，我們不能寫死合併的過程，具體怎麼合併應該由業務決定，由用戶去決定如何合併，所以合併的過程我們寫一個接口，具體的實現由用戶去實現

public interface Merger<E> {
    public E merge(E a, E b);
}

然後我們在構造方法中添加創建線段樹的過程(爲了創建線段樹，增加了一個輔助方法)

private Merger<E> merger;
//merger由用戶傳入 用戶決定如何合併
public SegmentTree(E[] arr, Merger<E> merger) {
    this.merger = merger;
    
    data = (E[]) new Object[arr.length];
    for (int i = 0; i < arr.length; i++) {
        data[i] = arr[i];
    }
    
    tree = (E[]) new Object[4 * data.length];
    //構造線段樹 創建根節點爲0，範圍爲[0,data.length - 1]的線段樹
    buildSegmentTree(0, 0, data.length - 1);
}

//在treeIndex創建一棵[l,r]的線段樹
private void buildSegmentTree(int treeIndex, int l, int r) {
    if (l == r) {
        tree[treeIndex] = data[l];
        return;
    }
    
    //l != r 那麼就要創建子樹的線段樹
    int leftTreeIndex = leftChild(treeIndex);
    int rightTreeIndex = rightChild(treeIndex);
    int mid = l + (r - l) / 2; //(l +r) / 2中l + r可能會大於int表示的範圍從而溢出
    buildSegmentTree(leftTreeIndex, l, mid);
    buildSegmentTree(rightTreeIndex, mid + 1, r);
    
    //融合的方法由用戶傳入
    tree[treeIndex] = merger.merge(tree[leftTreeIndex],tree[rightTreeIndex]);
}

爲了方便我們打印出線段樹，我們實現一個toString()方法

@Override
public String toString() {
    StringBuilder res = new StringBuilder();
    res.append("[");
    for (int i = 0; i < tree.length; i++) {
        if (tree[i] != null) {
            res.append(tree[i]);
        } else {
            res.append("null");
        }
        if (i != tree.length - 1) {
            res.append(", ");
        }
    }
    res.append("]");
    return res.toString();
}

查詢

實現代碼

public E query(int queryL, int queryR) {
    if (queryL < 0 || queryL >= data.length
            || queryR < 0 || queryR >= data.length
            || queryL > queryR) {
        throw new IllegalArgumentException("參數錯誤");
    }
    return query(0, 0, data.length - 1, queryL, queryR);
}
private E query(int treeIndex, int l, int r, int queryL, int queryR) {
    if (l == queryL && r == queryR) {
        return tree[treeIndex];
    }
    int leftChildIndex = leftChild(treeIndex);
    int rightChildIndex = rightChild(treeIndex);
    int mid = l + (r - l) / 2;
    if (queryL >= mid + 1) {
        return query(rightChildIndex, mid+1, r, queryL, queryR);
    } else if (queryR <= mid) {
        return query(leftChildIndex, l, mid, queryL, queryR);
    }
    E leftResult = query(leftChildIndex, l, mid, queryL, mid);
    E rightResult = query(rightChildIndex, mid + 1, r, mid + 1, queryR);
    return merger.merge(leftResult, rightResult);
}

更新

public void set(int index, E e) {
    if (index < 0 || index >= data.length) {
        throw new IllegalArgumentException("參數錯誤");
    }
    
    set(0, 0, data.length - 1, index, e);
}
private void set(int treeIndex, int l, int r, int index, E e) {
    if (l == r) {
        tree[treeIndex] = e;
        return;
    }
    int leftChildIndex = leftChild(treeIndex);
    int rightChildIndex = rightChild(treeIndex);
    int mid = l + (r - l) / 2;
    if (index >= mid + 1) {
        set(rightChildIndex, mid+1, r, index, e);
    } else {
        set(leftChildIndex, l, mid, index, e);
    }
    
    tree[treeIndex] = merger.merge(tree[leftChildIndex], tree[rightChildIndex]);
}

完整代碼

public class SegmentTree<E>{
    private E[] tree;
    private E[] data;
    private Merger<E> merger;

    public SegmentTree(E[] arr, Merger<E> merger) {
        this.merger = merger;

        data = (E[]) new Object[arr.length];
        for (int i = 0; i < arr.length; i++) {
            data[i] = arr[i];
        }

        tree = (E[]) new Object[4 * data.length];
        buildSegmentTree(0, 0, data.length - 1);
    }

    //在treeIndex創建一棵[l,r]的線段樹
    private void buildSegmentTree(int treeIndex, int l, int r) {
        if (l == r) {
            tree[treeIndex] = data[l];
            return;
        }

        int leftTreeIndex = leftChild(treeIndex);
        int rightTreeIndex = rightChild(treeIndex);

        int mid = l + (r - l) / 2; //(l +r) / 2中l + r可能會大於int表示的範圍從而溢出
        buildSegmentTree(leftTreeIndex, l, mid);
        buildSegmentTree(rightTreeIndex, mid + 1, r);

        tree[treeIndex] = merger.merge(tree[leftTreeIndex],tree[rightTreeIndex]);
    }

    public E query(int queryL, int queryR) {
        if (queryL < 0 || queryL >= data.length
                || queryR < 0 || queryR >= data.length
                || queryL > queryR) {
            throw new IllegalArgumentException("參數錯誤");
        }

        return query(0, 0, data.length - 1, queryL, queryR);
    }

    private E query(int treeIndex, int l, int r, int queryL, int queryR) {
        if (l == queryL && r == queryR) {
            return tree[treeIndex];
        }

        int leftChildIndex = leftChild(treeIndex);
        int rightChildIndex = rightChild(treeIndex);
        int mid = l + (r - l) / 2;

        if (queryL >= mid + 1) {
            return query(rightChildIndex, mid+1, r, queryL, queryR);
        } else if (queryR <= mid) {
            return query(leftChildIndex, l, mid, queryL, queryR);
        }

        E leftResult = query(leftChildIndex, l, mid, queryL, mid);
        E rightResult = query(rightChildIndex, mid + 1, r, mid + 1, queryR);

        return merger.merge(leftResult, rightResult);
    }


    public void set(int index, E e) {
        if (index < 0 || index >= data.length) {
            throw new IllegalArgumentException("參數錯誤");
        }

        set(0, 0, data.length - 1, index, e);
    }

    private void set(int treeIndex, int l, int r, int index, E e) {
        if (l == r) {
            tree[treeIndex] = e;
            return;
        }
        int leftChildIndex = leftChild(treeIndex);
        int rightChildIndex = rightChild(treeIndex);
        int mid = l + (r - l) / 2;
        if (index >= mid + 1) {
            set(rightChildIndex, mid+1, r, index, e);
        } else {
            set(leftChildIndex, l, mid, index, e);
        }

        tree[treeIndex] = merger.merge(tree[leftChildIndex], tree[rightChildIndex]);
    }

    public int getSize() {
        return data.length;
    }
    public E get(int index) {
        if (index < 0 || index >= data.length) {
            throw new IllegalArgumentException("參數錯誤");
        }
        return data[index];
    }

    private int leftChild(int index) {
        return 2 * index + 1;
    }
    private int rightChild(int index) {
        return 2 * index + 2;
    }

    @Override
    public String toString() {
        StringBuilder res = new StringBuilder();
        res.append("[");
        for (int i = 0; i < tree.length; i++) {
            if (tree[i] != null) {
                res.append(tree[i]);
            } else {
                res.append("null");
            }

            if (i != tree.length - 1) {
                res.append(", ");
            }
        }
        res.append("]");
        return res.toString();
    }
}