http://bbs.sjtu.edu.cn/bbstcon,board,Algorithm,reid,1225812893.html
我总结了一下,归纳如下: 1.1 SvS and Swapping SvS Algorithm 1 Pseudo-code for SvS SvS(set, k) 1: Sort the sets by size (|set[0]| ≤ |set[1]| ≤ . . . ≤ |set[k]|). 2: Let the smallest set s[0] be the candidate answer set. 3: for each set s[i], i = 1. . . k do initialize _[k] = 0. 4: for each set s[i], i = 1. . . k do 5: for each element e in the candidate answer set do 6: search for e in s[i] in the range l[i] to |s[i]|, 7: and update l[i] to the last position probed in the previous step. 8: if e was not found then 9: remove e from candidate answer set, 10: and advance e to the next element in the answer set. 11: end if 12: end for 13: end for 这是常用的一种算法,它首先是找出最短的两个集合,依次查找第一个集合里的元素是否 出现在第二个集合内部;Demaine考虑的Swapping_SvS和上述算法有稍微的不同,即是在每 次比较后,取包含更少元素的集合来与再下一个集合进行比较,这种算法在第一个集合和 第二个集合比较之后第二个集合反而更少的情况下效果更好,但实验表明这种情况并不多 见。 1.2 Small Adaptive Algorithm 2 Pseudo-code for Small_Adaptive Small_Adaptive(set, k) 1: while no set is empty do 2: Sort the sets by increasing number of remaining elements. 3: Pick an eliminator e = set[0][0] from the smallest set. 4: elimset ← 1. 5: repeat 6: search for e in set[elimset]. 7: increment elimset; 8: until s = k or e is not found in set[elimset] 9: if s = k then 10: add e to answer. 11: end if 12: end while 这是一种混合算法,结合了Svs和Adaptive的优点。它的特点是对每个集合按未被检查过的 元素个数进行排序,从中挑出未被检查过的元素个数最少和次少的集合进行比较,找到公 有的一个元素后,再在其他集合中进行查找,有某个集合查找完毕即结束。 1.3 Sequential and Random Sequential Algorithm 3 Pseudo-code for Sequential Sequential(set, k) 1: Choose an eliminator e = set[0][0], in the set elimset ← 0. 2: Consider the first set, i ← 1. 3: while the eliminator e _= ∞do 4: search in set[i] for e 5: if the search found e then 6: increase the occurrence counter. 7: if the value of occurrence counter is k then output e end if 8: end if 9: if the value of the occurrence counter is k, or e was not found then /*若计数到k或者e没有被找到*/ 10: update the eliminator to e ← set[i][succ(e)]. /*将e赋值为现在集合中下一个值*/ 11: end if 12: Consider the next set in cyclic order i ← i + 1 mod k. /*循环移位地选择新的集合*/ 13: end while Barbay and Kenyon引入的,对不确定复杂度的样本查找比较好,每次在各个集合中的查找 是用快速查找。 RSequential与Sequential的区别是Sequential挑选循环中下一个集合作为下一个搜索集合 ,而RSequential则是随机挑选一个集合。 1.4 Baeza-Yates and Baeza-Yates Sorted Algorithm 4 Pseudo-code for BaezaYates BaezaYates(set, k) 1: Sort the sets by size (|set[0]| ≤ |set[1]| ≤ . . . ≤ |set[k]|). 2: Let the smallest set set[0] be the candidate answer set. 3: for each set set[i], i = 1. . . k do 4: candidate ← BYintersect(candidate, set[i], 0, |candidate| ? 1, 0,|set[i]| ? 1) 5: sort the candidate set. 6: end for BYintersect(setA, setB, minA, maxA, minB, maxB) 1: if setA or setB are empty then return endif. 2: Let m = (minA + maxA)/2 and let medianA be the element at setA[m]. 3: Search for medianA in setB. 4: if medianA was found then 5: add medianA to result. 6: end if 7: Let r be the insertion rank of medianA in setB. 8: Solve the intersection recursively on both sides of r and m in each set. Baeza-Yates(巴伊赞-耶茨,他著有著名书籍《现代信息检索》)提出的方法,主要是利用 了分治思想,取出较短集合中的中间元素,在较长集合中搜索该元素,于是将较短和较长 集合均分为了2部分,在这2各部分中再递归搜索下去即可。注意:这样每次搜索完2个集合 ,输出的交集是无序的,因此需要将此交集再排序后,再和第3个集合进行比较搜索。 Baeza-Yates Sorted是对上述方法进行了改进,即在保存公有的元素时是按序保存的,保 存整段中间元素时必须保证前半段搜索到的中部元素已经被保存了,这样处理可以节省最 后将搜索到的交集再次排序的时间,但代价是中间处理的时候需要增加处理的细节。 1.5 总结 上面所有的算法最坏情况下都有线性的时间复杂度。BaezaYates、So_BaezaYates, Small _Adaptive和SvS在集合的大小不同时有显著优势,并且Small_Adaptive是惟一一个在算法 去除集合中元素导致集合的大小动态变化时,有更大的优势;Sequential and RSequenti al 对集合大小不敏感。
时间: 2024-10-08 12:44:05