解題區/解題區 - Zerojudge

b411. 記憶中的序列 2

contents

  1. 1. Problem
  2. 2. Sample Input
  3. 3. Sample Output
  4. 4. Solution
    1. 4.1. 另解

Problem

動態區間 K 小，有四種操作

  • 插入元素
  • 刪除元素
  • 修改元素
  • 版本回溯
  • 詢問某個區間的第 K 小元素

Sample Input

1
2
3
4
5
6
7
8
5
1 2 3 4 5
5
4 1 5 2
3 0 3
6 3 7 0
1 0
5 0 4 3

Sample Output

1
2
3
2
1
2

Solution

每次操作最多增加 $O(\sqrt{n})$ 記憶體，需要針對節點作抽象化，修改節點時再進行實體化，如此一來不用針對搜索路徑上的每一節點都進行複製，一般持久化平衡樹進行時，會直接複製整個節點，但塊狀鏈表若是複製下去會退化成時間、空間都是 $O(n)$ 的情況。

詢問時仍然需要二分搜索答案，看答案是否符合 K 小元素，也就是說要得到某個元素在區間中的排名。

塊狀鏈表代碼很醜，看看就好。如果需要支持修改、增加元素的區間 K 小，雖然複雜度高，但常數小、記憶體快取優勢，速度不比二元樹類型的資料結構來得慢，不管需不需要持久化塊狀鏈表都是這類型不錯的選擇。

另解

假設詢問為 $Q$ 個，聽說可以用區間 $[1, 2NQ]$ 的線段樹之類的超大區間線段樹，必要的探訪時進行開花 (將子節點打開)，利用標記計算相對位置來進行查詢。相對位置計算大概會在邊緣情況崩潰。

至於柳柳州提到的替罪羊樹是樹套樹下去持久化，每一個節點表示一個區間，接著在節點中建造一個平衡樹來查找區間 K 小元素。這樣的做法一樣需要二分搜索答案。但塊狀鏈表在速度跟記憶體都贏了，因為樹套樹的常數比較大，增加的記憶體也多。

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#include <stdio.h>
#include <math.h>
#include <stack>
#include <assert.h>
#include <vector>
#include <limits.h>
#include <map>
#include <algorithm>
using namespace std;
const int MAXN = 32767;
const int MAXQ = 131072;
class UnrolledLinkedList {
public:
    struct Node {
        vector<int> A, B;
        Node *next, *alias;
        Node() {
            A.clear(), B.clear(), next = alias = NULL;
        }
        Node* real() {
            if (alias)	return alias;
            return this;
        }
        void insert(int x, int val) {
            if (alias) {
                A = alias->A, B = alias->B;
                alias = NULL;
            }
            A.insert(A.begin() + x, val);
            B.resize(A.size());
            B[A.size()-1] = val;
            for (int i = (int) A.size()-1; i > 0 && B[i] < B[i-1]; i--)
                swap(B[i], B[i-1]);
        }
        void erase(int x) {
            int val = A[x];
            A.erase(A.begin()+x);
            B.erase(lower_bound(B.begin(), B.end(), val));
        }
        void change(int x, int val) {
            int t = A[x];
            A[x] = val;
            B.erase(lower_bound(B.begin(), B.end(), t));
            B.resize(A.size());
            B[A.size()-1] = val;
            for (int i = (int) A.size()-1; i > 0 && B[i] < B[i-1]; i--)
                swap(B[i], B[i-1]);
        }
        void relax() {
            B = A;
            sort(B.begin(), B.end());
        }
    };
    int PSIZE;
    stack<Node*> mem;
    Node* init(int A[], int n) { // A[1:n]
        free();
        PSIZE = max((int) sqrt(n), 2);
        for (int i = 1; ; i <<= 1) {
            if (i >= PSIZE) {
                PSIZE = i;
                break;
            }
        }
        
        Node *u, *v, *head;
        
        head = newNode();
        u = head, v = NULL;
        for (int i = 1; i <= n; i++) {
            if (u->A.size() == PSIZE) {
                u->next = newNode();
                v = u, u = u->next;
            }
            u->A.push_back(A[i]);
        }
        for (u = head; u != NULL; u = u->next)
            u->relax();
        return head;
    }
    Node* newNode() {
        Node* p = new Node();
        mem.push(p);
        return p;
    }
    Node* cloneNode(Node *u) {
        Node *t = u->real();
        Node *p = new Node();
        *p = *t, p->next = u->next, mem.push(p);
        return p; 
    }
    Node* aliasNode(Node *u) {
        Node *t = u->real();
        Node* p = new Node();
        p->alias = t, p->next = u->next, mem.push(p);
        return p;
    }
    Node* erase(Node *ver, int x) {
        Node *a, *b, *u, *v, *t;
        u = ver, a = NULL;
        for (int l = 1, r; u; l = r+1, u = u->next) {
            r = l + u->real()->A.size() - 1;
            if (l <= x && x <= r) {
                t = cloneNode(u);
                if (a == NULL)
                    a = t, b = a;
                else
                    b->next = t, b = b->next;
                t->erase(x - l);
                t->next = u->next;
                break;
            } else {
                if (a == NULL)
                    a = aliasNode(u), b = a;
                else
                    b->next = aliasNode(u), b = b->next;
            }
        }
        return relaxList(a);
    }
    Node* insert(Node *ver, int x, int val) {
        Node *a, *b, *u, *v, *t;
        u = ver, a = NULL;
        for (int l = 1, r; u; l = r+1, u = u->next) {
            r = l + u->real()->A.size() - 1;
            if (l-1 <= x && x <= r) {
                t = cloneNode(u);
                if (a == NULL)
                    a = t, b = a;
                else
                    b->next = t, b = b->next;
                t->insert(x - l + 1, val);
                t->next = u->next;
                break;
            } else {
                if (a == NULL)
                    a = aliasNode(u), b = a;
                else
                    b->next = aliasNode(u), b = b->next;
            }
        }
        return relaxList(a);
    }
    Node* change(Node *ver, int x, int val) {
        Node *a, *b, *u, *v, *t;
        u = ver, a = NULL;
        for (int l = 1, r; u; l = r+1, u = u->next) {
            r = l + u->real()->A.size() - 1;
            if (l <= x && x <= r) {
                t = cloneNode(u);
                if (a == NULL)
                    a = t, b = a;
                else
                    b->next = t, b = b->next;
                t->change(x - l, val);
                t->next = u->next;
                break;
            } else {
                if (a == NULL)
                    a = aliasNode(u), b = a;
                else
                    b->next = aliasNode(u), b = b->next;
            }
        }
        return a;
    }
    int findRank(Node *ver, int L, int R, int val, int threshold = INT_MAX) {
        Node *u, *v;
        int ret = 0;
        u = ver;
        for (int l = 1, r; u != NULL; u = u->next, l = r+1) {
            r = l + u->real()->A.size() - 1;
            if (L <= l && r <= R) {
                ret += upper_bound(u->real()->B.begin(), u->real()->B.end(), val) - u->real()->B.begin();
                L = r + 1;
            } else if ((l <= L && L <= r) || (l <= R && R <= r)) {
                int i = L - l;
                while (i < u->real()->A.size() && L <= R) {
                    if (u->real()->A[i] <= val)
                        ret++;
                    i++, L++;
                }
            }
            if (ret >= threshold)	return ret;
        }
        return ret;
    }
    int find(Node *ver, int L, int R, int k)  {	// kth-number
        int l = 0, r = 100000, mid, t = 0; // value in A[]
        while (l <= r) {
            mid = (l + r)/2;
            if (findRank(ver, L, R, mid, k) >= k)
                r = mid - 1, t = mid;
            else
                l = mid + 1;
        }
        return t;
    }
    Node* mergeNode(Node *u, Node *v) {
        Node *p, *q;
        Node *a = newNode();
        p = u->real(), q = v->real();
        a->next = v->next;
        a->A.insert(a->A.end(), p->A.begin(), p->A.end());
        a->A.insert(a->A.end(), q->A.begin(), q->A.end());
        a->B.resize(a->A.size());
        merge(p->B.begin(), p->B.end(), q->B.begin(), q->B.end(), a->B.begin());
        return a;
    }
    Node* splitNode(Node *u) {
        Node *t = u->real();
        Node *a = newNode(), *b = newNode();
        int n = t->A.size()/2;
        a->next = b, b->next = u->next;
        a->A.insert(a->A.begin(), t->A.begin(), t->A.begin()+n);
        b->A.insert(b->A.begin(), t->A.begin()+n, t->A.end());
        a->relax(), b->relax();
        return a;
    }
    Node* relaxList(Node *ver) {
        Node *a, *b, *u, *v, *t;
        int need = 0;
        for (u = ver, a = NULL; !need && u != NULL; u = u->next) {
            if (u->next != NULL && u->real()->A.size() + u->next->real()->A.size() < (PSIZE<<1)) // merge
            	need = 1;
            if (u->real()->A.size() > (PSIZE<<1))	// split
            	need = 1;
        }
        if (!need)
        	return ver;
        	
        for (u = ver, a = NULL; u != NULL; u = u->next) {
            if (u->next != NULL && u->real()->A.size() + u->next->real()->A.size() < (PSIZE<<1)) {	// merge
            	if (a == NULL)
            		a = mergeNode(u, u->next), b = a;
            	else
            		b->next = mergeNode(u, u->next), b = b->next;
                u = u->next;
            } else if (u->real()->A.size() > (PSIZE<<1)) {	// split
            	if (a == NULL)
            		a = splitNode(u), b = a->next;
            	else
            		b->next = splitNode(u), b = b->next->next;
            } else {
            	if (a == NULL)	
                    a = aliasNode(u), b = a;
            	else			
                    b->next = aliasNode(u), b = b->next;
            }
        }
        return a;
    }
    void debug(Node *head) {
        Node *u = head;
        printf("[");
    	while (u != NULL) {
        	for(int i = 0; i < u->real()->A.size(); i++)
            	printf("%d%s", u->real()->A[i], i != u->real()->A.size()-1 ? " " : " ");
        	u = u->next;
    	}
    	puts("]");
    }
    void free() {
        while (!mem.empty()) {
            Node *u = mem.top();
            mem.pop();
            delete u;
        }
    }
} dream; 
int A[MAXN], n, q;
UnrolledLinkedList::Node *ver[MAXQ];
int main() {
    int x, y, v, cmd;
    while (scanf("%d", &n) == 1) {
        for (int i = 1; i <= n; i++)
            scanf("%d", &A[i]);
        ver[0] = dream.init(A, n);
//		dream.debug(ver[0]);
        int encrypt = 0;
        scanf("%d", &q);
        for (int i = 1; i <= q; i++) {
            scanf("%d", &cmd);
            cmd ^= encrypt;
            if (cmd == 0) {			// back version v
                scanf("%d", &v);
                v ^= encrypt;
                ver[i] = ver[v];
            } else if (cmd == 1) { 	// insert A[x] = v
                scanf("%d %d", &x, &v);
                x ^= encrypt, v ^= encrypt;
                ver[i] = dream.insert(ver[i-1], x, v);
            } else if (cmd == 2) {	// delete A[x]
                scanf("%d", &x);
                x ^= encrypt;
                ver[i] = dream.erase(ver[i-1], x);
            } else if (cmd == 3) {	// make A[x] = v
                scanf("%d %d", &x, &v);
                x ^= encrypt, v ^= encrypt;
                ver[i] = dream.change(ver[i-1], x, v);
            } else if (cmd == 4) {
                scanf("%d %d %d", &x, &y, &v);
                x ^= encrypt, y ^= encrypt, v ^= encrypt;
                int t = dream.find(ver[i-1], x, y, v);
                printf("%d\n", t);
                encrypt = t;
                ver[i] = ver[i-1];
            } else {
                puts("WTF");
                return 0;
            }
//			dream.debug(ver[i]);
        }
    }
    return 0;
}