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Problem solution · Python

Beautiful Pairs

Beautiful Pairs: a Python solution using segment tree or range structure. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Segment tree or range structure
Source
Kamyu LeetCode Solutions
Length
277 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Segment tree or range structure

For Beautiful Pairs, the implementation stores interval information in a range-query data structure so updates and queries avoid rescanning the full input.

  1. Choose the aggregate stored for each interval or prefix.
  2. Build or initialize the structure from the input.
  3. Apply updates and combine the affected nodes to answer each query.

Code notes

  • 277 lines of Python from the credited upstream file beautiful-pairs.py.
  • The implementation visibly relies on sequence storage, ordered lookup.
  • No explicit loop blocks detected.

Complexity

Count the build once, then multiply the logarithmic update or query path by the number of operations.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from Kamyu LeetCode Solutions by kamyu104 and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeBeautiful Pairs · PythonPython
Use this to learn the idea, then write your own version.
# Time:  O(n) on average# Space: O(n) import randomimport itertoolsimport math  # random algorithms, variant of closest pair# reference: https://github.com/jilljenn/tryalgo/blob/master/tryalgo/closest_points.pyrandom.seed(0)class Solution(object):    def beautifulPair(self, nums1, nums2):        """        :type nums1: List[int]        :type nums2: List[int]        :rtype: List[int]        """        INF = float("inf")        def dist(a, b):            if a[2] > b[2]:                a, b = b, a            return [abs(a[0]-b[0])+abs(a[1]-b[1]), a[2], b[2]]         def cell(point, size):            x, y, _ = point            return math.floor(x/size), math.floor(y/size)         def improve():            lookup = {}            for p in points:                i, j = map(int, cell(p, result[0]/2.0))                for ni in xrange(i-2, (i+2)+1):                    for nj in xrange(j-2, (j+2)+1):                        if (ni, nj) not in lookup:                            continue                        d = dist(p, lookup[ni, nj])                        if d < result:                            result[:] = d                            return True                lookup[i, j] = p            return False         points = [(i, j, idx) for idx, (i, j) in enumerate(itertools.izip(nums1, nums2))]        result = [INF]*3        lookup = {}        for i in reversed(xrange(len(points))):            if points[i][:2] in lookup:                result = [0, i, lookup[points[i][:2]]]            lookup[points[i][:2]] = i        if result[0] == 0:            return result[1:]        random.shuffle(points)        result = dist(points[0], points[1])        while improve():            pass        return result[1:]  # Time:  O(nlogn)# Space: O(n)import itertools  # divide and conquer, merge sort, variant of closest pair# reference: https://www.baeldung.com/cs/minimal-manhattan-distanceclass Solution2(object):    def beautifulPair(self, nums1, nums2):        """        :type nums1: List[int]        :type nums2: List[int]        :rtype: List[int]        """        INF = float("inf")        MAX_NEIGHBOR_COUNT = (8+2)//2        def dist(a, b):            if a > b:                a, b = b, a            return [abs(points[a][0]-points[b][0])+abs(points[a][1]-points[b][1]), a, b]         def merge_sort(left, right):            def update(arr, i):  # added                for j in reversed(xrange(len(arr))):                    if points[i][1]-points[arr[j]][1] > result[0]:                        break                    result[:] = min(result, dist(i, arr[j]))                else:                    j = -1                assert((len(arr)-1)-j <= MAX_NEIGHBOR_COUNT)             if left == right:                return            mid = left+(right-left)//2            x = points[order[mid]][0]  # added            merge_sort(left, mid)            merge_sort(mid+1, right)            tmp, tmp_l, tmp_r = [], [], []            l, r = left, mid+1            while l <= mid or r <= right:                if r == right+1 or (l <= mid and points[order[l]][1] <= points[order[r]][1]):  # modified                    update(tmp_r, order[l])                    if x-points[order[l]][0] <= result[0]:  # added                        tmp_l.append(order[l])                    tmp.append(order[l])                    l += 1                else:                    update(tmp_l, order[r])                    if points[order[r]][0]-x <= result[0]:  # added                        tmp_r.append(order[r])                    tmp.append(order[r])                    r += 1            order[left:right+1] = tmp         points = [(i, j) for i, j in itertools.izip(nums1, nums2)]        result = [INF]*3        lookup = {}        for i in reversed(xrange(len(points))):            if points[i] in lookup:                result = [0, (i, lookup[points[i]])]            lookup[points[i]] = i        if result[0] == 0:            return result[1]        order = range(len(points))        order.sort(key=lambda x: points[x][0])        merge_sort(0, len(points)-1)        return result[1:]  # Time:  O(nlogn)# Space: O(n)import itertools  # divide and conquer, merge sort, variant of closest pair# reference: https://www.baeldung.com/cs/minimal-manhattan-distanceclass Solution3(object):    def beautifulPair(self, nums1, nums2):        """        :type nums1: List[int]        :type nums2: List[int]        :rtype: List[int]        """        INF = float("inf")        MAX_NEIGHBOR_COUNT = 8        def dist(a, b):            if a > b:                a, b = b, a            return [abs(points[a][0]-points[b][0])+abs(points[a][1]-points[b][1]), a, b]         def merge_sort(left, right):            if left == right:                return            mid = left + (right-left)//2            x = points[order[mid]][0]  # added            merge_sort(left, mid)            merge_sort(mid+1, right)            r = mid+1            tmp = []            for l in xrange(left, mid+1):                while r <= right and points[order[r]][1] < points[order[l]][1]:  # modified                    tmp.append(order[r])                    r += 1                tmp.append(order[l])            order[left:left+len(tmp)] = tmp             # added below            stripe = [order[i] for i in xrange(left, right+1) if abs(points[order[i]][0]-x) <= result[0]]            for i in xrange(len(stripe)-1):                for j in xrange(i+1, len(stripe)):                    x, y = stripe[i], stripe[j]                    if points[y][1]-points[x][1] > result[0]:                        break                    result[:] = min(result, dist(x, y))                else:                    j = len(stripe)                assert(j-(i+1) <= MAX_NEIGHBOR_COUNT)         points = [(i, j) for i, j in itertools.izip(nums1, nums2)]        result = [INF]*3        lookup = {}        for i in reversed(xrange(len(points))):            if points[i] in lookup:                result = [0, (i, lookup[points[i]])]            lookup[points[i]] = i        if result[0] == 0:            return result[1]        order = range(len(points))        order.sort(key=lambda x: points[x][0])        merge_sort(0, len(points)-1)        return result[1:]  # Time:  O(nlogn)# Space: O(n)import itertools  # segment treeclass Solution4(object):    def beautifulPair(self, nums1, nums2):        """        :type nums1: List[int]        :type nums2: List[int]        :rtype: List[int]        """        INF = float("inf")        # Range Maximum Query        class SegmentTree(object):            def __init__(self, N,                         build_fn=lambda _: [-INF, -INF],  # modified                         query_fn=lambda x, y: y if x is None else x if y is None else max(x, y),                         update_fn=lambda x: x):                self.tree = [None]*(2*2**((N-1).bit_length()))                self.base = len(self.tree)//2                self.query_fn = query_fn                self.update_fn = update_fn                for i in xrange(self.base, self.base+N):                    self.tree[i] = build_fn(i-self.base)                for i in reversed(xrange(1, self.base)):                    self.tree[i] = query_fn(self.tree[2*i], self.tree[2*i+1])             def update(self, i, h):                x = self.base+i                self.tree[x] = self.update_fn(h)                while x > 1:                    x //= 2                    self.tree[x] = self.query_fn(self.tree[x*2], self.tree[x*2+1])             def query(self, L, R):                if L > R:                    return [-INF, -INF]  # modified                L += self.base                R += self.base                left = right = None                while L <= R:                    if L & 1:                        left = self.query_fn(left, self.tree[L])                        L += 1                    if R & 1 == 0:                        right = self.query_fn(self.tree[R], right)                        R -= 1                    L //= 2                    R //= 2                return self.query_fn(left, right)         def dist(a, b):            if a > b:                a, b = b, a            return [abs(points[a][0]-points[b][0])+abs(points[a][1]-points[b][1]), a, b]         points = [(i, j) for i, j in itertools.izip(nums1, nums2)]        result = [INF]*3        lookup = {}        for i in reversed(xrange(len(points))):            if points[i] in lookup:                result = [0, (i, lookup[points[i]])]            lookup[points[i]] = i        if result[0] == 0:            return result[1]        order = range(len(points))        order.sort(key=lambda x: points[x][0])        y_set = set(y for _, y in points)        y_to_idx = {y:i for i, y in enumerate(sorted(y_set))}        st1, st2 = SegmentTree(len(y_to_idx)), SegmentTree(len(y_to_idx))        for i in order:            j = -st1.query(0, y_to_idx[points[i][1]]-1)[1]  # min((xi-xj)+(yi-yj) for j in range(y_to_idx[points[i][1])) = (xi+yi)-max((xj+yj) for j in range(y_to_idx[points[i][1]))            if j != INF:                assert(points[j][1] < points[i][1])                result = min(result, dist(i, j))            st1.update(y_to_idx[points[i][1]], [points[i][0]+points[i][1], -i])            j = -st2.query(y_to_idx[points[i][1]], len(y_to_idx)-1)[1]  # min((xi-xj)+(yj-yi) for j in range(y_to_idx[points[i][1], len(y_to_idx))) = (xi-yi)-max((xj-yj) for j in range(y_to_idx[points[i][1], len(y_to_idx))            if j != INF:                assert(points[j][1] >= points[i][1])                result = min(result, dist(i, j))            st2.update(y_to_idx[points[i][1]], [points[i][0]-points[i][1], -i])        return result[1:]

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