Boyer–Moore majority vote algorithm

https://en.wikipedia.org/wiki/Boyer–Moore_majority_vote_algorithm

https://leetcode.com/problems/online-majority-element-in-subarray/

这个算法是用来选择“一个序列中超过一半的元素”. 这个算法我很早在一本数据结构和算法的书中就学习到过, 但是作者给出的算法和BM算法真的一点都不像。所以我当时只是记住了怎么使用,但是没有深入理解这个算法有什么变形。


当时作者给出的算法是这样的:不断地消解两个连续的元素,如果他们一致就保留两个,否则全部丢弃。但是这个算法会对数组修改。

def majority_vote(a):
    while len(a) != 1:
        k = 0
        for i in range(len(a), 2):
            if a[i] == a[i+1]:
                a[k] = a[i]
                k += 1
        if len(a) % 2 != 0:
            a[k] = a[-1]
            k += 1
        if k == 0:
            return None
        a = a[:k]
    return a[0]

print(majority_vote([1,2,1,1,2]))
print(majority_vote([1,2,1,1,2,2]))

而wiki给出的BM算法更加简单,只需要维护几个变量就可以了。

def majority_vote(a):
    p, c = None, 0
    for x in a:
        if c == 0:
            p = x
            c = 1
        elif x == p:
            c += 1
        else:
            c -= 1
    if c == 0: return None
    return p

print(majority_vote([1,2,1,1,2]))
print(majority_vote([1,2,1,1,2,2]))


如果如果对第一个算法稍加改动,结合BM算法思想的话,就可以对任意区间求解majority.

我们将区间用区间树来表示,节点有两个值 `freq`(表示值出现频次)和 `value` (表示具体什么值). 我们在处理节点a, b的时候

只有当freq>0的话,val才是majority元素.

class SegTree:
    def __init__(self):
        self.val = self.freq = 0
        self.left = self.right = None
        self.start = self.end = 0


def bm_vote(a: SegTree, b: SegTree, c: SegTree):
    if a.val == b.val:
        c.val = a.val
        c.freq = a.freq + b.freq
        return

    if b.freq < a.freq:
        a, b = b, a

    c.val = b.val
    c.freq = b.freq - a.freq
    return


def build_seg_tree(arr, s, e):
    if s > e:
        return None
    if s == e:
        t = SegTree()
        t.val = arr[s]
        t.freq = 1
        t.start = s
        t.end = e
        return t

    m = (s + e) // 2
    left = build_seg_tree(arr, s, m)
    right = build_seg_tree(arr, m + 1, e)
    t = SegTree()
    t.start = s
    t.end = e
    t.left = left
    t.right = right
    bm_vote(left, right, t)
    return t


def query_seg_tree(t, s, e):
    if s <= t.start and t.end <= e:
        return t
    m = (t.start + t.end) // 2
    if (m + 1) > e:
        # 只搜索左边
        x = query_seg_tree(t.left, s, e)
    elif m < s:
        x = query_seg_tree(t.right, s, e)
    else:
        a = query_seg_tree(t.left, s, m)
        b = query_seg_tree(t.right, m + 1, e)
        x = SegTree()
        bm_vote(a, b, x)
    return x