## Sunday, November 13, 2016

### LeetCode 456 - 132 Pattern

Given a sequence of n integers a1, a2, ..., an, a 132 pattern is a subsequence ai, aj, ak such that i < j < k and ai < ak < aj. Design an algorithm that takes a list of n numbers as input and checks whether there is a 132 pattern in the list.
Note: n will be less than 15,000.
Example 1:
Input: [1, 2, 3, 4]

Output: False

Explanation: There is no 132 pattern in the sequence.

Example 2:
Input: [3, 1, 4, 2]

Output: True

Explanation: There is a 132 pattern in the sequence: [1, 4, 2].

Example 3:
Input: [-1, 3, 2, 0]

Output: True

Explanation: There are three 132 patterns in the sequence: [-1, 3, 2], [-1, 3, 0] and [-1, 2, 0].
BST（Binary Search Tree 二叉查找树）
首先利用TreeMap（采用红黑树实现）统计nums中所有元素的个数，记为tm

将num在tm中的计数-1，计数为0时将num从tm中删去

如果num < min，则更新min值为num

否则，若tm中大于min的最小元素<num，则返回true

public boolean find132pattern(int[] nums) { TreeMap<Integer, Integer> tm = new TreeMap<>(); for (int num : nums) { tm.put(num, tm.getOrDefault(num, 0) + 1); } int min = Integer.MAX_VALUE; for (int num : nums) { int cnt = tm.get(num); if (cnt > 1) { tm.put(num, cnt - 1); } else { tm.remove(num); } if (num <= min) { min = num; } else { Integer target = tm.higherKey(min); if (target != null && target < num) { return true; } } } return false; }
X.
https://leetcode.com/articles/132-pattern/#approach-5-using-binary-search-accepted
    public boolean find132pattern(int[] nums) {
if (nums.length < 3)
return false;
int[] min = new int[nums.length];
min[0] = nums[0];
for (int i = 1; i < nums.length; i++)
min[i] = Math.min(min[i - 1], nums[i]);
for (int j = nums.length - 1, k = nums.length; j >= 0; j--) {
if (nums[j] > min[j]) {
k = Arrays.binarySearch(nums, k, nums.length, min[j] + 1);
if (k < 0)
k = -1 - k;
if (k < nums.length && nums[k] < nums[j])
return true;
nums[--k] = nums[j];
}
}
return false;
}

X. Use Stack
http://blog.csdn.net/mebiuw/article/details/53193012
public boolean find132pattern(int[] nums) { Stack<Range> stack = new Stack<>(); for(int num : nums) { Range cur = new Range(num, num); while(!stack.isEmpty() && cur.max > stack.peek().min) { cur.min = Math.min(stack.peek().min, cur.min); cur.max = Math.max(stack.peek().max, cur.max); stack.pop(); } stack.push(cur); if(stack.peek().min < num && num < stack.peek().max) return true; } return false; }
https://discuss.leetcode.com/topic/68193/java-o-n-solution-using-stack-in-detail-explanation
https://discuss.leetcode.com/topic/67816/o-n-time-o-n-space-java-solution-using-stack-13ms
You are absolutely right because intuition doesn't lead us to Stack solution for this problem.
At the very beginning, I tried to use two variables, i.e. minmax, to denote the min-max intervals, and update them as I encounter new numbers, but I soon found that only two variables cannot easily maintain the position information (as you update min and max, the previous ones lost).
So we need to keep "multiple min-max pairs" here. Then what can we use? We can use a List, which is definitely feasible. And what is the advantage of List? That is we can randomly access any position, but as I go along my thinking process to figure out how to maintain/merge different intervals, I found that we only need to fetch those intervals from the List sequentially, so List seems to be too "powerful" at this time, and Stack is a perfect LIFO container.
BTW, for all Stack questions, we can always use List, but Stack makes it easier to code and maintain

The idea is that we can use a stack to keep track of previous min-max intervals.
Here is the principle to maintain the stack:
For each number num in the array
If stack is empty:
• push a new Pair of num into stack
If stack is not empty:
• if num < stack.peek().min, push a new Pair of num into stack
• if num > stack.peek().min, we first pop() out the peek element, denoted as last
• if num < last.max, we are done, return true;
• if num > last.max, we merge num into last, which means last.max = num.
Once we update last, if stack is empty, we just push back last.
However, the crucial part is:
If stack is not empty, the updated last might:
• Entirely covered stack.peek(), i.e. last.min < stack.peek().min (which is always true) && last.max > stack.peek().max, in which case we keep popping out stack.peek().
• Form a 1-3-2 pattern, we are done ,return true
So at any time in the stack, non-overlapping Pairs are formed in descending order by their min value, which means the min value of peek element in the stack is always the min value globally.
i = 6nums = [ 9, 11, 8, 9, 10, 7, 9 ], S1 candidate = 9S3 candidate = NoneStack = Empty
i = 5nums = [ 9, 11, 8, 9, 10, 7, 9 ], S1 candidate = 7S3 candidate = NoneStack = [9]
i = 4nums = [ 9, 11, 8, 9, 10, 7, 9 ], S1 candidate = 10S3 candidate = NoneStack = [9,7]
i = 3nums = [ 9, 11, 8, 9, 10, 7, 9 ], S1 candidate = 9S3 candidate = 9Stack = [10]
i = 2nums = [ 9, 11, 8, 9, 10, 7, 9 ], S1 candidate = 8S3 candidate = 9Stack = [10,9] We have 8<9, sequence found!
   class Pair{
int min, max;
public Pair(int min, int max){
this.min = min;
this.max = max;
}
}
public boolean find132pattern(int[] nums) {
Stack<Pair> stack = new Stack();
for(int n: nums){
if(stack.isEmpty() || n <stack.peek().min ) stack.push(new Pair(n,n));
else if(n > stack.peek().min){
Pair last = stack.pop();
if(n < last.max) return true;
else {
last.max = n;
while(!stack.isEmpty() && n >= stack.peek().max) stack.pop();
// At this time, n < stack.peek().max (if stack not empty)
if(!stack.isEmpty() && stack.peek().min < n) return true;
stack.push(last);
}

}
}
return false;
}
https://discuss.leetcode.com/topic/67881/single-pass-c-o-n-space-and-time-solution-8-lines-with-detailed-explanation
We want to search for a sub sequence (s1,s2,s3)
INTUITION: The problem would be simple if we want to find sequence with s1 > s2 > s3, we just need to find s1, followed by s2 and s3. Now if we want to find a 132 sequence, we need to switch up the order of searching. we want to first find s2, followed by s3, then s1.
IDEA: We can start from either side but I think starting from the end allow us to finish in a single pass. The idea is to start from end and search for a candidate for s2 and s3. A number becomes a candidate for s3 if there is any number on the left (s2) that is bigger than it.
DETECTION: Keep track of the largest candidate of s3 and once we encounter any number smaller than s3, we know we found a valid sequence since s1 < s3 implies s1 < s2.
IMPLEMENTATION:
1. Have a stack, each time we store a new number, we first pop out all numbers that are smaller than that number. The numbers that are popped out becomes candidate for s3.
2. We keep track of the maximum of such s3.
3. Once we encounter any number smaller than s3, we know we found a valid sequence since s1 < s3 implies s1 < s2.
RUNTIME: Each item is pushed and popped once at most, the time complexity is therefore O(n).
    bool find132pattern(vector<int>& nums) {
int s3 = INT_MIN;
stack<int> st;
for( int i = nums.size()-1; i >= 0; i -- ){
if( nums[i] < s3 ) return true;
else while( !st.empty() && nums[i] > st.top() ){
s3 = max( s3, st.top() ); st.pop();
}
st.push(nums[i]);
}
return false;
}
public boolean find132pattern(int[] nums) {
Stack<Integer> stack = new Stack<>();
for (int i = nums.length - 1, two = Integer.MIN_VALUE; i >= 0; stack.push(nums[i--]))
if (nums[i] < two) return true;
else for (; !stack.empty() && nums[i] > stack.peek(); two = Math.max(two, stack.pop()));
return false;
}
https://segmentfault.com/a/1190000007507137

Class Pair {
int min;
int max;
public Pair(int min, int max) {
this.min = min;
this.max = max;
}
}

public boolean find123Pattern(int[] nums) {
if(nums == null || nums.length < 3) return false;
Pair cur = new Pair(nums[0], nums[0]);
Stack<Pair> stack = new Stack<>();
for(int i = 1; i < nums.length; i ++) {
if(nums[i] < cur.min) {
stack.push(cur);
cur = new Pair(nums[i], nums[i]);
}
else if(nums[i] > cur.max) {
while(!stack.isEmpty() && stack.peek().max <= nums[i]) {
stack.pop();
}
if(!stack.isEmpty() && stack.peek.max > nums[i]) {
return true;
}
cur.max = nums[i];
}
else if(nums[i] > cur.min && nums[i] < cur.max) {
return true;
}
}
return false;
}
X.
https://discuss.leetcode.com/topic/67615/share-my-easy-and-simple-solution
the worst case time complexity for this one is O(n^2) when the array is in ascending order, right?
Idea: Find peak and bottom
For every [bottom, peak], find if there is one number bottom<number<peak.
    public boolean find132pattern(int[] nums) {
if(nums.length<3) return false;
Integer low = null, high = null;
int start = 0, end = 0;
while(start<nums.length-1){
while(start<nums.length-1 && nums[start]>=nums[start+1]) start++;
// start is lowest now
int m = start+1; //no need to use m - use end instead
while(m<nums.length-1 && nums[m]<=nums[m+1]) m++;
// m is highest now
int j = m+1;
while(j<nums.length){
if(nums[j]>nums[start] && nums[j]<nums[m]) return true;
j++;
}
start = m+1;
}
return false;
}
X.
https://discuss.leetcode.com/topic/67592/java-straightforward
public boolean find132pattern(int[] nums) {
int len = nums.length;
if(len < 3) return false;
int[] max_cache = new int[len];
int[] min_cache = new int[len];

min_cache[0] = nums[0];
for(int j = 1; j < len ; j++){
min_cache[j] = Math.min(min_cache[j-1], nums[j]);
}
max_cache[len-1] = nums[len-1];
for(int j = len-2; j >= 0; j--){
max_cache[j] = Math.max(max_cache[j+1], nums[j]);
}
for(int i = 1; i < len-1; i++){
int val = nums[i];
int left = min_cache[i-1];
if(max_cache[i+1] > left && val > max_cache[i+1]) return true;
for(int j = i+1; j < len; j++){
if(nums[j] > left && val > nums[j]) return true;
}
}
return false;
}
X. brute force
https://discuss.leetcode.com/topic/68242/java-solutions-from-o-n-3-to-o-n-for-132-pattern
I. Naive O(n^3) solution
simply check every (i, j, k) combination to see if there is any 132 pattern.
public boolean find132pattern(int[] nums) {
for (int i = 0; i < nums.length; i++) {
for (int j = i + 1; j < nums.length; j++) {
for (int k = j + 1; k < nums.length; k++) {
if (nums[i] < nums[k] && nums[k] < nums[j]) return true;
}
}
}
return false;
}
II. Improved O(n^2) solution
To reduce the time complexity down to O(n^2), we need to do some observations. In the naive solution above, let's assume we have index j fixed, what should index i be so that it is most probable we will have a 132 pattern? Or in other words, what should i be so that we will be certain there is no such 132 pattern for combination (*, j, *) whenever there is no 132 pattern for combination of (i, j, *)? (Here * means any index before or after index j.)
The answer lies in the fact that once the first two numbers nums[i] and nums[j] are fixed, we are up to find the third number nums[k] which will be within the range (nums[i], nums[j]) (the two bounds are exclusive). Intuitively the larger the range is, the more likely there will be a number "falling into" it. Therefore we need to choose index i which will maximize the range (nums[i], nums[j]). Since the upper bound nums[j] is fixed, this is equivalent to minimizing the lower bound nums[i]. Thus it is clear ishould be the index of the minimum element of the subarray nums[0, j) (left inclusive, right exclusive).
Since we are scanning index j from the beginning of the input array nums, we can keep track of the minimum element of the subarray from index 0 up to j - 1 without rescanning it. Therefore the first two loops in the naive solution can be combined into one and leads to the following O(n^2) solution:
public boolean find132pattern(int[] nums) {
for (int j = 0, min = Integer.MAX_VALUE; j < nums.length; j++) {
min = Math.min(nums[j], min);
if (min == nums[j]) continue;

for (int k = nums.length - 1; k > j; k--) {
if (min < nums[k] && nums[k] < nums[j]) return true;
}
}

return false;
}

While this solution can be accepted, it runs slow. One obvious drawback is that in the second loop we are throwing away information about elements in the right part of nums that may be "useful" for later combinations. It turns out we can retain this "useful" information by applying the classic space-time tradeoff, which leads to the following O(n) time and O(n) space solution.
https://leetcode.com/articles/132-pattern/#approach-5-using-binary-search-accepted
Now, assume that we have somehow found a $nums[i],nums[j]$ pair. Our task now reduces to finding out a $nums[k]$($Kk>j>i)$, which falls in the range $(nums[i], nums[j])$. Now, to maximize the likelihood of a $nums[k]$ falling in this range, we need to increase this range as much as possible. Since, we started off by fixing a $nums[j]$, the only option in our hand is to choose a minimum value of $nums[i]$ given a particular $nums[j]$. Once, this pair $nums[i],nums[j]$, has been found out, we simply need to traverse beyond the index $j$ to find if a $nums[k]$ exists for this pair satisfying the 132 criteria.
Based on the above observations, while traversing over the $nums$ array choosing various values of $nums[j]$, we simultaneously keep a track of the minimum element found so far(excluding $nums[j]$). This minimum element always serves as the $nums[i]$ for the current $nums[j]$. Thus, we only need to traverse beyond the $j^{th}$ index to check the $nums[k]$'s to determine if any of them satisfies the 132 criteria.

III. Optimized O(n) solution
As I mentioned, to further reduce the time complexity, we need to record the "useful" information about elements in the right part of the input array nums. Since these elements are located at the right part of nums, it will be hard to do so if we are scanning the array from the beginning. So the idea is to scan it from the end while in the meantime keep track of the "useful" information. But still at each index j, we need to know the minimum element for subarray nums[0, j). This can be done by doing a pre-scan in the forward direction and memorize the results for each index in an auxiliary array (we will call the array as arr whose element arr[j] will denote the minimum element in the subarray nums[0, j)).
Until now we are kinda vague about the exact meaning of "useful" information, so let's try to be more specific. Assume we're currently scanning (from the end) the element with index j, our task is to find two elements nums[i] and nums[k] to determine if there exists a 132 pattern, with i < j < k. The left element nums[i], as it has been shown in part II, will be chosen as arr[j], the minimum element of subarray nums[0, j). What about the right element nums[k]?
The answer to that will address the meaning of "useful" information. First note we are only interested in elements that are greater than arr[j], so it is sensible to maintain only those elements. Second, among all these qualified elements, which one will be the most probable to fall into the range (nums[i], nums[j])? I would say it is the smallest one (i.e., if the smallest one is out of the range, all others will also be out of range). So to sum up, the "useful" information for current index j will be a collection of scanned elements that are greater than arr[j], and nums[k] will be chosen as the smallest one if the collection is not empty.
From the analyses above, it looks like we have to do some sorting stuff for the retained elements (or at least find a way to figure out its smallest element). Well, it turns out these elements will be sorted automatically due to the fact that arr[j'] >= arr[j] as long as j' < j. Here is how it goes, which is a proof by induction.
At the beginning we have an empty collection and of course it is sorted. Now suppose we are at index j and the corresponding collection is still sorted, let's see if it remains so at index j - 1. First we will check if nums[j] is greater than arr[j]. If not, we simply continue to j - 1. Since the collection is intact so it will be sorted at j - 1. Otherwise, we need to remove elements in the collection that are no greater than arr[j] (this is necessary because some smaller elements may be left over in the collection from previous steps). After removal, we then compare the first element in the collection with nums[j] to see if a 132 pattern has been found, provided the collection is not empty. If so, return true. Otherwise one of the following must be true: the collection is empty or nums[j] is no greater than the first element in the collection. In either case the collection is sorted. Now if we have arr[j - 1] < nums[j], we need to add nums[j] to the collection since it is a qualified number for arr[j - 1]. Again in either case the collection will remain sorted after addition (if it is empty, after addition there is only one element; otherwise since the added element is no greater than the first element in the collection before addition, it will become the new first element after addition and the collection stays sorted).
Here is the program with O(n) time and space complexity. There is one minor optimization based on the observation that the total number of elements in the collection will never exceed the total number of elements scanned so far. Therefore the right part of the arrarray can be used to serve as the collection. For time complexity, each element in the input array nums will be pushed into and popped out from the collection (or stack to be exact) at most once, the time complexity will be O(n) despite of the nested loop.
public boolean find132pattern(int[] nums) {
int[] arr = Arrays.copyOf(nums, nums.length);

for (int i = 1; i < nums.length; i++) {
arr[i] = Math.min(nums[i - 1], arr[i - 1]);
}

for (int j = nums.length - 1, top = nums.length; j >= 0; j--) {
if (nums[j] <= arr[j]) continue;
while (top < nums.length && arr[top] <= arr[j]) top++;
if (top < nums.length && nums[j] > arr[top]) return true;
arr[--top] = nums[j];
}

return false;
}
https://t.du9l.com/2016/11/leetcode-456-132-pattern/

bool find132pattern(vector<int>& nums) {
if (nums.empty()) return false;
int n = nums.size();
vector<int> min1(n);
min1[0] = nums[0];
for (int i = 1; i < n; ++i) {
min1[i] = min(min1[i-1], nums[i]);
}
typedef pair<int, int> pii;
stack<pii> s;
vector<int> max1;
for (int i = 0; i < n; ++i) {
int I = nums[i];
while (!s.empty() && s.top().first <= I) s.pop();
if (s.empty()) {
max1.push_back(-1);
} else {
max1.push_back(s.top().second);
}
s.push(make_pair(I, i));
}
for (int i = n-1; i >= 0; --i) {
int I = nums[i];
int j = max1[i];
if (j == -1 || j == 0) continue;
if (min1[j-1] < I) return true;
}
return false;
}

discuss中有一些更为精炼的方法。一些也是求范围的；另外一些是求出一个数左边的最小，再看右边有没有比左边最小大且比自己小的数