USACO 2022 US Open Contest, Platinum Problem 1. 262144 Revisited

USACO 2022 US Open Contest, Platinum Problem 1. 262144 Revisited

Bessie likes downloading games to play on her cell phone, even though she does find the small touch screen rather cumbersome to use with her large hooves.

She is particularly intrigued by the current game she is playing. The game starts with a sequence of NN positive integers a1,a2,,aNa1,a2,…,aN (2N262,1442≤N≤262,144), each in the range 11061…106. In one move, Bessie can take two adjacent numbers and replace them with a single number equal to one greater than the maximum of the two (e.g., she might replace an adjacent pair (5,7)(5,7) with an 88). The game ends after N1N−1 moves, at which point only a single number remains. The goal is to minimize this final number.

Bessie knows that this game is too easy for you. So your job is not just to play the game optimally on aa, but for every contiguous subsequence of aa.

Output the sum of the minimum possible final numbers over all N(N+1)2N(N+1)2 contiguous subsequences of aa.

 

INPUT FORMAT (input arrives from the terminal / stdin):

First line contains NN.The next line contains NN space-separated integers denoting the input sequence.

 

OUTPUT FORMAT (print output to the terminal / stdout):

A single line containing the sum.

 

SAMPLE INPUT:

6
1 3 1 2 1 10

SAMPLE OUTPUT:

115

There are 672=216⋅72=21 contiguous subsequences in total. For example, the minimum possible final number for the contiguous subsequence [1,3,1,2,1][1,3,1,2,1] is 55, which can be obtained via the following sequence of operations:

 

original    -> [1,3,1,2,1]
combine 1&3 -> [4,1,2,1]
combine 2&1 -> [4,1,3]
combine 1&3 -> [4,4]
combine 4&4 -> [5]

Here are the minimum possible final numbers for each contiguous subsequence:

 

final(1:1) = 1
final(1:2) = 4
final(1:3) = 5
final(1:4) = 5
final(1:5) = 5
final(1:6) = 11
final(2:2) = 3
final(2:3) = 4
final(2:4) = 4
final(2:5) = 5
final(2:6) = 11
final(3:3) = 1
final(3:4) = 3
final(3:5) = 4
final(3:6) = 11
final(4:4) = 2
final(4:5) = 3
final(4:6) = 11
final(5:5) = 1
final(5:6) = 11
final(6:6) = 10


SCORING:

 

  • Test cases 2-3 satisfy N300N≤300.
  • Test cases 4-5 satisfy N3000N≤3000.
  • In test cases 6-8, all values are at most 4040.
  • In test cases 9-11, the input sequence is non-decreasing.
  • Test cases 12-23 satisfy no additional constraints.

Problem credits: Benjamin Qi

USACO 2022 US Open Contest, Platinum Problem 1. 262144 Revisited 题解(翰林国际教育提供,仅供参考)

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(Analysis by Benjamin Qi)

I will refer to contiguous sequences as intervals. Define the value of an interval to be the minimum possible final number it can be converted into.

Subtask 1: Similar to 248, we can apply dynamic programming on ranges. Specifically, if dp[i][j]dp[i][j] denotes the value of interval iji…j, then

 

dp[i][j]={Aaiminik<jmax(dp[i][k],dp[k+1][j])+1i=ji<j.dp[i][j]={Aaii=jmini≤k<jmax(dp[i][k],dp[k+1][j])+1i<j.

The time complexity is O(N3)O(N3).

 

#include <bits/stdc++.h>
using namespace std;

template <class T> using V = vector<T>;
#define all(x) begin(x), end(x)

template <class T> void ckmin(T &a, const T &b) { a = min(a, b); }

int main() {
	ios::sync_with_stdio(false);
	cin.tie(nullptr);
	int N;
	cin >> N;
	V<int> A(N);
	for (int &x : A) cin >> x;
	V<V<int>> dp(N, V<int>(N));
	for (int i = N - 1; i >= 0; --i) {
		dp.at(i).at(i) = A.at(i);
		for (int j = i + 1; j < N; ++j) {
			dp[i][j] = INT_MAX;
			for (int k = i; k < j; ++k) {
				ckmin(dp.at(i).at(j),
					  max(dp.at(i).at(k), dp.at(k + 1).at(j)) + 1);
			}
		}
	}
	int64_t ans = 0;
	for (int i = 0; i < N; ++i) {
		for (int j = i; j < N; ++j) {
			ans += dp.at(i).at(j);
		}
	}
	cout << ans << "\n";
}

Subtask 2: We can optimize the solution above by more quickly finding the maximum kk′ such that dp[i][k]dp[k+1][j]dp[i][k′]≤dp[k′+1][j]. Then we only need to consider k{k,k+1}k∈{k′,k′+1} when computing dp[i][j]dp[i][j]. Using the observation that kk′ does not decrease as jj increases and ii is held fixed leads to a solution in O(N2)O(N2):

 

#include <bits/stdc++.h>
using namespace std;

template <class T> using V = vector<T>;
#define all(x) begin(x), end(x)

template <class T> void ckmin(T &a, const T &b) { a = min(a, b); }

int main() {
	ios::sync_with_stdio(false);
	cin.tie(nullptr);
	int N;
	cin >> N;
	V<int> A(N);
	for (int &x : A) cin >> x;
	V<V<int>> dp(N, V<int>(N));
	for (int i = N - 1; i >= 0; --i) {
		dp.at(i).at(i) = A.at(i);
		int k = i - 1;
		for (int j = i + 1; j < N; ++j) {
			while (k + 1 < j && dp.at(i).at(k + 1) <= dp.at(k + 2).at(j)) ++k;
			dp[i][j] = INT_MAX;
			ckmin(dp[i][j], dp.at(k + 1).at(j));
			ckmin(dp[i][j], dp.at(i).at(k + 1));
			++dp[i][j];
		}
	}
	int64_t ans = 0;
	for (int i = 0; i < N; ++i) {
		for (int j = i; j < N; ++j) {
			ans += dp.at(i).at(j);
		}
	}
	cout << ans << "\n";
}

Alternatively, finding kk′ with binary search leads to a solution in O(N2logN)O(N2log⁡N).

Subtask 3: Similar to 262144, we can use binary lifting.

For each of v=1,2,3,v=1,2,3,…, I count the number of intervals with value at least vv. The answer is the sum of this quantity over all vv.

 

#include <bits/stdc++.h>
using namespace std;

template <class T> using V = vector<T>;
#define all(x) begin(x), end(x)

int main() {
	ios::sync_with_stdio(false);
	cin.tie(nullptr);
	int N;
	cin >> N;
	V<int> A(N);
	for (int &x : A) cin >> x;
	V<V<int>> with_val;
	for (int i = 0; i < N; ++i) {
		while (size(with_val) <= A[i]) with_val.emplace_back();
		with_val.at(A[i]).push_back(i);
	}
	V<int> nex(N + 1);
	iota(all(nex), 0);
	int64_t ans = 0;
	for (int v = 1;; ++v) {
		if (nex[0] == N) {
			cout << ans << "\n";
			exit(0);
		}
		// add all intervals with value >= v
		for (int i = 0; i <= N; ++i) ans += N - nex[i];
		for (int i = 0; i <= N; ++i) nex[i] = nex[nex[i]];
		if (v < size(with_val)) {
			for (int i : with_val.at(v)) {
				nex[i] = i + 1;
			}
		}
	}
}

Subtask 4: For each ii from 11 to NN in increasing order, consider the values of all intervals with right endpoint ii. Note that the value vv of each such interval must satisfy v[Ai,Ai+log2i]v∈[Ai,Ai+⌈log2⁡i⌉] due to AA being sorted. Thus, it suffices to be able to compute for each vv the minimum ll such that dp[l][i]vdp[l][i]≤v. To do this, we maintain a partition of 1i1…i into contiguous subsequences such that every contiguous subsequence has value at most AiAi and is leftwards-maximal (extending any subsequence one element to the left would cause its value to exceed AiAi). When transitioning from i1i−1 to ii, we merge every two consecutive contiguous subsequences AiAi1Ai−Ai−1 times and then add contiguous subsequence [i,i][i,i] to the end of the partition.

 

#include <bits/stdc++.h>
using namespace std;

template <class T> using V = vector<T>;
#define all(x) begin(x), end(x)

int main() {
	ios::sync_with_stdio(false);
	cin.tie(nullptr);
	int N;
	cin >> N;
	vector<int> A(N);
	for (int &x : A) cin >> x;
	assert(is_sorted(all(A)));
	// left endpoints of each partition interval in decreasing order
	deque<int> left_ends;
	int64_t ans = 0;
	for (int i = 0; i < N; ++i) {
		if (i) {
			for (int v = A[i - 1]; v < A[i]; ++v) {
				if (size(left_ends) == 1) break;
				// merge every two consecutive intervals in partition
				deque<int> n_left_ends;
				for (int j = 0; j < (int)size(left_ends); ++j) {
					if ((j & 1) || j + 1 == (int)size(left_ends)) {
						n_left_ends.push_back(left_ends[j]);
					}
				}
				swap(left_ends, n_left_ends);
			}
		}
		left_ends.push_front(i); // add [i,i] to partition
		int L = i + 1;
		for (int v = A[i]; L; ++v) {
			int next_L = left_ends.at(
			    min((int)size(left_ends) - 1, (1 << (v - A[i])) - 1));
			ans += (int64_t)(L - next_L) * v;
			L = next_L;
		}
	}
	cout << ans << "\n";
}

Full Credit: Call an interval relevant if it is not possible to extend it to the left or to the right without increasing its value.

Claim: The number of relevant intervals is O(NlogN)O(Nlog⁡N).

Proof: See the end of the analysis.

We'll compute the same quantities as in subtask 3, but this time, we'll transition from v1v−1 to vv in time proportional to the number of relevant intervals with value v1v−1 plus the number of relevant intervals with value vv, this will give us a solution in O(NlogN)O(Nlog⁡N).

For a fixed value of vv, say that an interval [l,r)[l,r) is a segment if it contains no value greater than vv, and min(Al1,Ar)>vmin(Al−1,Ar)>v. Say that an interval is maximal with respect to vv if it has value at most vv and extending it to the left or right would cause its value to exceed vv. Note that a maximal interval [l,r)[l,r) must be relevant, and it must either have value equal to vv or be a segment.

My code follows. ivals[i]ivals[i] stores all maximal intervals contained within the segment with left endpoint ii. The following steps are used to transition from v1v−1 to vv:

 

  1. Apply halvehalve on all segments containing more than one maximal interval (the left endpoints of every such segment are stored by activeactive). Before, all intervals within the segment were maximal with respect to v1v−1. After, all intervals within the segment are maximal with respect to vv.
  2. Add a segment and a maximal interval of the form [i,i+1)[i,i+1) for each ii satisfying Ai=vAi=v, and then merge adjacent segments.

 

#include <bits/stdc++.h>
using namespace std;

template <class T> using V = vector<T>;
#define all(x) begin(x), end(x)

int main() {
	ios::sync_with_stdio(false);
	cin.tie(nullptr);
	int N;
	cin >> N;
	V<int> A(N);
	V<V<int>> with_A;
	for (int i = 0; i < N; ++i) {
		cin >> A[i];
		while ((int)size(with_A) <= A[i]) with_A.emplace_back();
		with_A[A[i]].push_back(i);
	}
	// sum(l ... r)
	auto sum_arith = [&](int64_t l, int64_t r) {
		return (r + l) * (r - l + 1) / 2;
	};
	// total number of intervals covered by list of maximal intervals
	auto contrib = [&](const list<pair<int, int>> &L) {
		int64_t ret = 0;
		for (auto it = begin(L);; ++it) {
			auto [x, y] = *it;
			if (next(it) == end(L)) {
				ret += sum_arith(0, y - x);
				break;
			} else {
				int next_x = next(it)->first;
				ret += int64_t(next_x - x) * y - sum_arith(x, next_x - 1);
			}
		}
		return ret;
	};
	int64_t num_at_least = (int64_t)N * (N + 1) / 2;
	auto halve = [&](list<pair<int, int>> &L) {
		if (size(L) <= 1) return;
		num_at_least += contrib(L);
		int max_so_far = -1;
		list<pair<int, int>> n_L;
		auto it = begin(L);
		for (auto [x, y] : L) {
			while (next(it) != end(L) && next(it)->first <= y) ++it;
			if (it->second > max_so_far) {
				n_L.push_back({x, max_so_far = it->second});
			}
		}
		swap(L, n_L);
		num_at_least -= contrib(L);
	};

	// doubly linked list to maintain segments
	V<int> pre(N + 1);
	iota(all(pre), 0);
	V<int> nex = pre;

	int64_t ans = 0;
	V<int> active; // active segments
	// maximal intervals within each segment
	V<list<pair<int, int>>> ivals(N + 1);

	for (int v = 1; num_at_least; ++v) {
		ans += num_at_least; // # intervals with value >= v
		V<int> n_active;
		for (int l : active) {
			halve(ivals[l]);
			if (size(ivals[l]) > 1) n_active.push_back(l);
		}
		if (v < (int)size(with_A)) {
			for (int i : with_A[v]) {
				int l = pre[i], r = nex[i + 1];
				bool should_add = size(ivals[l]) <= 1;
				pre[i] = nex[i + 1] = -1;
				nex[l] = r, pre[r] = l;
				ivals[l].push_back({i, i + 1});
				--num_at_least;
				ivals[l].splice(end(ivals[l]), ivals[i + 1]);
				should_add &= size(ivals[l]) > 1;
				if (should_add) n_active.push_back(l);
			}
		}
		swap(active, n_active);
	}
	cout << ans << "\n";
}

Proof of Claim: Let f(N)f(N) denote the maximum possible number of relevant subarrays for a sequence of size NN. We can show that f(N)O(logN!)O(NlogN)f(N)≤O(log⁡N!)≤O(Nlog⁡N). This upper bound can be attained when all elements of the input sequence are equal.

The idea is to consider a Cartesian tree of aa. Specifically, suppose that one of the maximum elements of aa is located at position pp (1pN1≤p≤N). Then

 

f(N)f(p1)+f(Np)+#(relevant intervals containing p).f(N)≤f(p−1)+f(N−p)+#(relevant intervals containing p).

WLOG we may assume 2pN2p≤N.

Lemma:

 

#(relevant intervals containing p)O(plog(Np))#(relevant intervals containing p)≤O(plog⁡(Np))

Proof of Lemma: We can in fact show that

 

#(relevant intervals containing p with value ap+k)min(p,2k):0klog2N.#(relevant intervals containing p with value ap+k)≤min(p,2k):0≤k≤⌈log2⁡N⌉.

The pp comes from all relevant intervals with a fixed value having distinct left endpoints and the 2k2k comes from the fact that to generate a relevant interval of value ap+k+1ap+k+1 containing pp, you must start with a relevant interval of value ap+kap+k and choose to extend it either to the right or to the left.

To finish, observe that the summation log2Nk=0#(relevant intervals containing p with value ap+k)∑k=0⌈log2⁡N⌉#(relevant intervals containing p with value ap+k) is dominated by the terms satisfying log2pklog2Nlog2⁡p≤k≤⌈log2⁡N⌉

Since O(plogNp)O(log(Np))O(logN!(p1)!(Np)!)O(plog⁡Np)≤O(log⁡(Np))≤O(log⁡N!(p−1)!(N−p)!), the claim follows from the lemma by induction:

 

f(N)f(p1)+f(Np)+#(relevant intervals containing p)O(log(p1)!)+O(log(Np)!)+O(logN!log(p1)!log(Np)!)O(logN!).f(N)≤f(p−1)+f(N−p)+#(relevant intervals containing p)≤O(log⁡(p−1)!)+O(log⁡(N−p)!)+O(log⁡N!−log⁡(p−1)!−log⁡(N−p)!)≤O(log⁡N!).

Here is an alternative approach by Danny Mittal that uses both the idea from the subtask 4 solution and the Cartesian tree. He repeatedly finds the index of the rightmost maximum amidamid of the input sequence, solves the problem recursively on a1mid1a1…mid−1 and amid+1Namid+1…N, and then adds the contribution of all intervals containing amidamid. This also runs in O(NlogN)O(Nlog⁡N).

 

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.*;
 
public class Revisited262144Array {
    static int n;
    static int[] xs;
 
    static int[] left;
    static int[] right;
 
    static int[] forward;
    static int[] reverse;
 
    static long answer = 0;
 
    public static int reduceForward(int start, int length, int lgFactor) {
        if (lgFactor == 0) {
            return length;
        }
        int factor = 1 << Math.min(lgFactor, 20);
        int j = start;
        for (int k = start + factor - 1; k < start + length - 1; k += factor) {
            forward[j++] = forward[k];
        }
        forward[j++] = forward[start + length - 1];
        return j - start;
    }
 
    public static void reduceReverse(int start, int length, int lgFactor) {
        if (lgFactor == 0) {
            return;
        }
        int factor = 1 << Math.min(lgFactor, 20);
        if (length > factor) {
            int j = start + 1;
            for (int k = start + 1 + ((length - factor - 1) % factor); k < start + length; k += factor) {
                reverse[j++] = reverse[k];
            }
        }
    }
 
    public static int funStuff(int from, int mid, int to, int riseTo) {
        if (from > to) {
            return 0;
        }
        int leftLength = funStuff(from, left[mid], mid - 1, xs[mid]);
        int rightLength = funStuff(mid + 1, right[mid], to, xs[mid]);
        int leftStart = from;
        int rightStart = mid + 1;
        int last = mid - 1;
        for (int j = 1; j <= rightLength + 1; j++) {
            int frontier = j > 1 ? forward[rightStart + (j - 2)] : mid;
            long weight = frontier - last;
            last = frontier;
            int lastInside = mid + 1;
            int leftLast = leftLength == 0 ? mid : reverse[leftStart];
            for (int d = 0; d <= 18 && lastInside > leftLast; d++) {
                if (1 << d >= j) {
                    int frontierInside;
                    if (1 << d == j) {
                        frontierInside = mid;
                    } else if (1 << d <= j + leftLength) {
                        frontierInside = reverse[leftStart + leftLength + j - (1 << d)];
                    } else {
                        frontierInside = leftLast;
                    }
                    long weightInside = lastInside - frontierInside;
                    lastInside = frontierInside;
                    answer += weight * weightInside * ((long) (xs[mid] + d));
                }
            }
        }
        forward[leftStart + leftLength] = mid;
        System.arraycopy(forward, rightStart, forward, leftStart + leftLength + 1, rightLength);
        reverse[leftStart + leftLength] = mid;
        System.arraycopy(reverse, rightStart, reverse, leftStart + leftLength + 1, rightLength);
        int length = reduceForward(leftStart, leftLength + 1 + rightLength, riseTo - xs[mid]);
        reduceReverse(leftStart, leftLength + 1 + rightLength, riseTo - xs[mid]);
        return length;
    }
 
    public static void main(String[] args) throws IOException {
        BufferedReader in = new BufferedReader(new InputStreamReader(System.in));
        n = Integer.parseInt(in.readLine());
        StringTokenizer tokenizer = new StringTokenizer(in.readLine());
        xs = new int[n];
        for (int j = 0; j < n; j++) {
            xs[j] = Integer.parseInt(tokenizer.nextToken());
        }
        left = new int[n];
        right = new int[n];
        ArrayDeque<Integer> stack = new ArrayDeque<>();
        for (int j = 0; j < n; j++) {
            while (!stack.isEmpty() && xs[stack.peek()] <= xs[j]) {
                left[j] = stack.pop();
            }
            if (!stack.isEmpty()) {
                right[stack.peek()] = j;
            }
            stack.push(j);
        }
        while (stack.size() > 1) {
            stack.pop();
        }
        forward = new int[n];
        reverse = new int[n];
        funStuff(0, stack.pop(), n - 1, Integer.MAX_VALUE);
        System.out.println(answer);
    }
}

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