# Complexity of Union-Split-Find Problems. Katherine Jane Lai

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1 Complexity of Union-Split-Find Problems by Katherine Jane Lai S.B., Electrical Engineering and Computer Science, MIT, 2007 S.B., Mathematics, MIT, 2007 Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Electrical Engineering and Computer Science at the Massachusetts Institute of Technology June 2008 c 2008 Massachusetts Institute of Technology All rights reserved. Author Department of Electrical Engineering and Computer Science May 23, 2008 Certified by Erik D. Demaine Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by Arthur C. Smith Professor of Electrical Engineering Chairman, Department Committee on Graduate Theses

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3 Complexity of Union-Split-Find Problems by Katherine Jane Lai Submitted to the Department of Electrical Engineering and Computer Science on May 23, 2008, in partial fulfillment of the requirements for the Degree of Master of Engineering in Electrical Engineering and Computer Science Abstract In this thesis, we investigate various interpretations of the Union-Split-Find problem, an extension of the classic Union-Find problem. In the Union-Split-Find problem, we maintain disjoint sets of ordered elements subject to the operations of constructing singleton sets, merging two sets together, splitting a set by partitioning it around a specified value, and finding the set that contains a given element. The different interpretations of this problem arise from the different assumptions made regarding when sets can be merged and any special properties the sets may have. We define and analyze the Interval, Cyclic, Ordered, and General Union-Split-Find problems. Previous work implies optimal solutions to the Interval and Ordered Union-Split-Find problems and an Ω(log n/ log log n) lower bound for the Cyclic Union-Split-Find problem in the cell-probe model. We present a new data structure that achieves a matching upper bound of O(log n/ log log n) for Cyclic Union-Split- Find in the word RAM model. For General Union-Split-Find, no o(n) bound is known. We present a data structure which has an Ω(log 2 n) amortized lower bound in the worst case that we conjecture has polylogarithmic amortized performance. This thesis is the product of joint work with Erik Demaine. Thesis Supervisor: Erik D. Demaine Title: Associate Professor of Electrical Engineering and Computer Science 3

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5 Acknowledgments I would like to thank my thesis supervisor, Erik Demaine, for taking me on as his student and for our conversations that taught me much about research, teaching, academia, and life in general. I would also like to thank Mihai Pǎtraşcu for his contribution to this work. Thank you also to David Johnson, Timothy Abbott, Eric Price, and everyone who has patiently discussed these problems with me. Finally, I thank my family and friends for their support and their faith in my ability to do research. 5

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7 Contents 1 Introduction Models of Computation Interval Union-Split-Find Problem Cyclic Union-Split-Find Problem Ordered Union-Split-Find Problem General Union-Split-Find Problem Known Lower and Upper Bounds Cyclic Union-Split-Find Problem Data Structure A Balanced-Parenthesis String Prefix Sums Array Summary Data Auxiliary Pointers Space Analysis Parent Operation Insertions and Deletions Incremental Updates Splits and Merges of B-Tree Nodes Proof of Tight Bounds Complexity General Union-Split-Find Problem Data Structure A Lower Bound

8 3.3 Attempted Upper Bounds Open Problems and Future Directions

9 List of Figures 1-1 Example of Interval Union-Split-Find Example of Cyclic Union-Split-Find Example of Ordered Union-Split-Find Balanced-Parenthesis String Stored at Internal Nodes Compressed Balanced-Parenthesis String Auxiliary Pointers Paths Taken By Parent Operation in the B-Tree Performing Parent in O(1) on a Machine Word A Level-Linked Tree Real Line Representation Pseudocode for General Union Operation Expensive Merge Sequence: Merge Tree Representation Expensive Merge Sequence: Real Line Representation Incorrect Potential Function: Sum of Interleaves Incorrect Potential Function: Sum of Pairwise Interleaves Counterexample: Merging Costliest Pair of Sets First Counterexample: Merging Smallest Interleaving Set First

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11 Chapter 1 Introduction While the Union-Find problem [6] is well-studied in the realm of computer science, we argue that there are several different but natural interpretations of an analogous Union- Split-Find problem. These different versions of the problem arise from making various assumptions regarding when sets can be merged and any special properties the sets may have. This thesis considers four natural interpretations of the Union-Split-Find problem. While tight bounds have previously been found for some versions of the Union-Split-Find problem, the complexity of other variations, especially the most general form of the problem, have not been well studied. This thesis is the product of joint work with Erik Demaine. The Union-Find problem operates on disjoint sets of elements and requires three operations: make-set(x), union(x, y), and find(x). The make-set(x) operation constructs a new singleton set containing x. The union(x, y) operation merges the two sets of items that contain x and y, respectively, into a new set, destroying the two original sets in the process. Finally, the find(x) operation returns a canonical name for the set that contains x. Tarjan [20] showed that these operations can be supported in Θ(α(m, n)) amortized time, where α(m, n) is the inverse Ackermann function, and that this bound is optimal in the pointer-machine model. The inverse Ackermann function grows very slowly and for all practical purposes is at most 4 or 5. Fredman and Saks [10] showed a matching lower bound of Ω(α(m, n)) in the cell-probe model under the assumption that all machine words are of size O(log n). The General Union-Split-Find problem adds support for a split operation that divides a set of elements into two sets, in addition to the original make-set, union, and find operations. 11

12 Here we require that the elements come from an ordered universe so that the split operation has an intuitive and efficient meaning. Specifically, split(x, y) partitions the set that contains x into two new sets, the first containing the set of elements with value less than or equal to y, and the second containing the set of the elements with value greater than y. The original set that contained x is destroyed in the process. Despite the naturality of this general Union-Split-Find problem, it has not been studied before to our knowledge. Other, more restricted versions of the Union-Split-Find problem have been studied in the past. In this thesis, we define four natural variations: Interval, Cyclic, Ordered, and General Union-Split-Find. 1.1 Models of Computation To analyze the complexity of these problems and their algorithms, we primarily consider three different models of computation. The first model, the pointer-machine model [20], is the most restricted model we use. A pointer-machine data structure is represented by a directed graph of nodes each containing some constant number of integers and having a constant number of outgoing edges pointers to other nodes. Supported instructions in this model are generally of the form of storing data, making comparisons, and creating new nodes and pointers. The pointer machine is limited in that it is not able to perform arithmetic on the pointers. Since it is the weakest model we use, upper bounds in this model hold in the other two models as well. Second is the word Random Access Machine (RAM) model [1, 11], one of the more realistic models for the performance of present-day computers. In a word RAM, values are stored in an array of machine words. Under the standard transdichotomous assumption, machine words consist of w = Ω(log n) bits. The ith word in the array can be accessed (read or written) in O(1) time. The word RAM model also allows arithmetic and bitwise operations (and, or, bit shifts, etc.) on O(1) words in O(1) time. Thus the running time of an algorithm evaluated in the word RAM model is proportional to the number of memory accesses and the number of machine-word operations performed. Lastly, the cell-probe model [15, 22] is the strongest model of computation we consider. This model is like the word RAM model, except that all machine-word operations are free. The cost of an algorithm in the cell-probe model is just the number of memory accesses. 12

13 Because this model is strictly more powerful than all the previous models, any lower bound in this model applies to the other two models as well. 1.2 Interval Union-Split-Find Problem Mehlhorn, Näher, and Alt [14] are the only ones to previously define a problem called Union-Split-Find, but it is a highly specialized form which we call Interval Union-Split- Find to distinguish from the other variations under discussion. This problem is not the intuitive counterpart for Union-Find because it operates on intervals of values over a static universe of elements instead of arbitrary disjoint sets over a dynamic universe of elements. Only adjacent intervals can be unioned together, and there is no equivalent to the make-set operation required by the Union-Find problem. This problem is remarkable, however, in that the van Emde Boas priority queue [21] solves it optimally in Θ(log log n) time per operation. Mehlhorn et al. [14] show that the elements and their set membership can be modeled as a list of marked or unmarked elements where the marked elements indicate the left endpoints of the intervals, as shown in the example in Figure 1-1. The union and split operations then correspond to unmark(x) and mark(x), which unmark or mark the endpoint x between the two affected sets. The find(x) operation corresponds to finding the nearest marked element before or at x. This problem is thus equivalent to the classic dynamic predecessor problem over a linear universe, with the mark and unmark operations corresponding to inserting and deleting elements, and the find operation to the predecessor query. The van Emde Boas priority queue, as described by van Emde Boas, Kaas, and Zulstra [21], can perform the necessary operations and provides an upper bound of O(log log n) time per operation in the pointer-machine model. Intervals (the sets): Marked/Stored items: [1, 10) [10, 15) [15, 22) [22, n 3) [n 3, n) Indices of items: n 3 n Figure 1-1: Interval Union-Split-Find: Intervals in the ordered finite universe can be represented by storing the left endpoints in a van Emde Boas priority queue. In their analysis of the problem, Mehlhorn et al. [14] prove an Ω(log log n) lower bound for this problem in the pointer-machine model. Pǎtraşcu and Thorup [18] prove a stronger 13

14 result, showing that the van Emde Boas structure is optimal in the cell-probe model when the size of the universe is O(n), which is the case here. Thus we have a tight bound of Θ(log log n) time per operation for the complexity of Interval Union-Split-Find. 1.3 Cyclic Union-Split-Find Problem The Cyclic Union-Split-Find problem is a problem proposed by Demaine, Lubiw, and Munro in 1997 [8], as a special case of the dynamic planar point location problem. In the dynamic planar point location problem, we maintain a decomposition of the plane into polygons, or faces, subject to edge insertions, edge deletions, and queries that return the face containing a given query point. In the Cyclic Union-Split-Find problem, we support the same operations, but the faces are limited to a decomposition of a convex polygon divided by chords. Solving this computational geometry problem is equivalent to maintaining the set of points contained in each face of the polygon. Thus the union operation combines two adjacent faces by deleting a chord, and the split operation splits a face by inserting a chord. The find operation returns which face the given query point is in. Similar to Interval Union- Split-Find, Cyclic Union-Split-Find is a special case of General Union-Split-Find where restrictions are placed on which sets can be unioned together; for example, faces that are not touching cannot be unioned together with the deletion of a chord. To indicate a canonical name for each set or face, we note that each face except one can be represented by a unique chord in the graph. Given that a particular face is represented by some chord c i, if a new chord c j is inserted that splits this face, we continue to represent one of these new faces with the same chord c i and represent the other new face with the chord c j. The one face that is not represented by a chord stems from the initial state of a convex polygon with no chords and one face. We can transform the Cyclic Union-Split-Find problem into an equivalent problem by making a single cut at some point of the perimeter of the polygon and unrolling the polygon so that the problem becomes a one-dimensional line segment with paired endpoints preserved from the chords (see Figure 1-2 for an example). This graph can be represented by a balanced-parenthesis string as shown by Munro and Raman [16], with the two endpoints of each chord becoming the open and close symbols of a matching pair of parentheses. Balanced-parenthesis strings are strings of symbols ( and ) such that each ( symbol 14

15 B A L BK K J C D CJ GJ I E DF GI H A B C D E F G H I J K L F (a) G ()( ()( ()( ()) ()(( ()) ())) ()) (b) Figure 1-2: Cyclic Union-Split-Find: (a) A polygon with chords before it is unrolled; all faces except one can be uniquely represented by a neighboring chord. (b) The corresponding one-page graph and balanced-parenthesis string. The cut used to construct the one-page graph is located at the point L shown on the polygon in (a). can be paired with a ) symbol that appears after it and vice versa, and these pairs do not cross. Thus (()())() is a valid string while (()))( is not. Thus the corresponding operations in this balanced-parenthesis problem are the following: inserting pairs of parentheses (split), deleting pairs of parentheses (union), and finding the immediately enclosing parent pair of parentheses of a given query pair (find). We can observe that the Interval Union-Split-Find problem can be reduced to the Cyclic Union-Split-Find problem by representing the items and the boundaries of the sets, or intervals, as a balanced-parenthesis string. We represent each item as a pair of parentheses (), and sets are denoted by additional pairs of parentheses that enclose the interval of pairs of parentheses that represent the set s items. Thus the union operation is equivalent to first deleting the appropriate pairs of parentheses that denote the two adjacent sets to be merged, and then inserting the new pair of parentheses that appropriately contains the newly merged set. The split operation is the reverse of the union operation and involves deleting the original set s delimiters and inserting two pairs of parentheses that denote the two new smaller sets. The find operation is simply the parent operation that returns the parentheses that encloses the query item s parentheses. This reduction implies that there is at least a lower bound of Ω(log log n) for Cyclic Union-Split-Find in all three models under consideration. 15

16 If we translate the balanced-parenthesis string into an array of integers by replacing each open parenthesis with a value of +1 and each close parenthesis with a value of 1, the parent operation is equivalent to solving a predecessor problem on a signed prefix sum array. For example, if a query pair s open parenthesis has value 3 in the array of prefix sums for the balanced-parenthesis string, the open parenthesis of the parent pair is located at the first parenthesis to the left of the query parenthesis that has a value of 2. The signed prefix sum array must be dynamically maintained during insertions and deletions of parentheses. Husfeldt, Rauhe, and Skyum [12] proved a lower bound of Ω(log n/ log log n) for the dynamic signed prefix sum problem and for the planar point location problem in the cell-probe model, where the word sizes were assumed to be Θ(log n). The proof more generally establishes a lower bound of Ω(log w n) on Cyclic Union-Split-Find. It remained open whether this lower bound is tight. We develop a data structure in Chapter 2 that achieves a matching upper bound of O(log w n) time per operation in the word RAM model. 1.4 Ordered Union-Split-Find Problem Another interesting variation is the Ordered Union-Split-Find problem, or perhaps more appropriately, the Concatenate-Split-Find problem, where the problem operates on ordered lists or paths over a dynamic universe of elements. New elements can be created using the make-set operation. For this problem, union(x, y) can be performed only when all items in the set containing x are less than all the items in the set containing y. Merging two sets that lack this ordering property would require multiple split and union operations. See Figure 1-3 for an example. a) b) c) Figure 1-3: Ordered Union-Split-Find: Unlike Interval Union-Split-Find, sets of ordered lists with arbitrarily many interleaving values can be formed. List (b) in this example cannot be unioned with either of the other two because concatenation would not maintain the sorted order. However, list (a) can be unioned with list (c). 16

17 The Ordered Union-Split-Find problem generalizes both Interval Union-Split-Find and Cyclic Union-Split-Find. The Interval Union-Split-Find problem easily reduces to Ordered Union-Split-Find if we represent each interval or set as an actual ordered list of all its members. Two intervals in the Interval Union-Split-Find problem can be unioned only when they are adjacent to each other, so they can always be unioned by the union operation of Ordered Union-Split-Find. The split and find operations are trivially equivalent for the two problems. Similarly, the operations required by the Cyclic Union-Split-Find problem can be implemented using those of the Ordered Union-Split-Find problem. In the balanced-parenthesis form of the Cyclic Union-Split-Find problem, a set is a list of parenthesis pairs that share the same parent. As such, Ordered Union-Split-Find can be used to maintain those sets as paths. In the Ordered Union-Split-Find problem, the equivalent of inserting a pair of parentheses in the Cyclic Union-Split-Find problem is taking the ordered list that contains the set denoted by the new pair s parent, splitting it at the locations of the parenthesis insertions, and concatenating the two outer pieces of the original list (if they have nonzero length) and the new list together in order of appearance in the balanced-parenthesis string. This in effect splits off a new set formed out of a subsequence in the original set. A deletion of a pair of parentheses translates to the reverse operations: one split operation at the location where the child set will be inserted into the parent set, another split and deletion for the element corresponding to the pair of parentheses to be deleted, and up to two union operations to concatenate the three remaining sets together. Performing the parent operation is simply a find operation as before. Because the sets are ordered lists and can be thought of as paths, we can easily represent the Ordered Union-Split-Find problem as that of maintaining paths of elements that may be concatenated or split. The find operation is thus the same as returning the path containing the query element. Together, these operations support the problem of dynamic connectivity on disjoint paths. Dynamic connectivity is the problem of maintaining connected components of undirected graphs, subject to three operations: inserting edges, deleting edges, and testing whether two vertices are connected. Pǎtraşcu and Demaine [17] prove a lower bound of Ω(log n) on dynamic connectivity in the cell-probe model that applies even for graphs composed of only disjoint paths; this bound is therefore a lower bound on the Ordered Union-Split-Find problem. Link-cut trees, data structures first described by 17

18 Sleator and Tarjan [19], support operations on a forest of rooted unordered trees in O(log n) time. In particular, they support make-tree(), link(v, w), cut(v, w), and find root(v), which correspond to the Ordered Union-Split-Find operations of make-set, union, split, and find, respectively. In fact, because we only need to maintain paths rather than rooted unordered trees, we can also just use the special case of link-cut paths, which are the building blocks of link-cut trees. Link-cut trees and paths work in the pointer-machine model. Thus, in all three models, the upper and lower bounds for Ordered Union-Split-Find match, yielding a tight bound of Θ(log n) time. As discussed earlier, the Interval and Cyclic Union-Split-Find problems both reduce to the Ordered Union-Split-Find problem. The Ordered Union-Split-Find problem is not only a generalization of both those problems, but it is also a step closer to the most general version of the Union-Split-Find problem. It supports the make-set operation which allows the number of managed elements to get arbitrarily large without rebuilding the entire data structure. It also allows for the existence of different sets occupying the same interval of values as seen with the first two sets in Figure 1-3. It is not completely general, however, because the union operation can only operate between the end of one path and the beginning of another, and the split operation preserves the order of the paths. This can be useful for certain applications where order is meaningful and needs to be preserved, but it does not easily support the more general idea of splitting an arbitrary set on the basis of the elements associated values. 1.5 General Union-Split-Find Problem To our knowledge, the General Union-Split-Find problem has not explicitly been studied previously, even though it is the most natural interpretation of the Union-Split-Find problem. Like the Union-Find problem, it maintains arbitrary sets of disjoint elements. The sets can be created through any sequence of make-set(x), union(x, y), and split(x) operations, starting from the empty collection of sets. The make-set, union, and find operations function exactly as in the original Union-Find problem. The split operation splits a given set by value. The difference between Ordered and General Union-Split-Find is that the union operation now needs to be capable of merging two arbitrary disjoint sets of elements. Referring back to the example of Ordered Union-Split-Find in Figure 1-3, there is now no 18

19 restriction on which two sets can be unioned together even if the interval of values represented in one set overlaps completely with the interval of the other. All of the previously mentioned variations of the Union-Split-Find problem can be reduced to this generalized problem, so it too has a lower bound of Ω(log n) per operation. There is also clearly a trivial O(n) upper bound per operation: simply store a set s elements in an unordered linked list. The union operation is then simply a concatenation of the two sets lists in O(1) time, and make-set is still an O(1) time operation. The split operation can be performed in O(n) time by simply partitioning a set around the value used for the split via a linear scan through the set s items, and the find operation can also be performed in O(n) time by scanning an entire set s list for its minimum element. While the General Union-Split-Find problem may not have been explicitly studied before, others have studied the related problem of merging and splitting sets of items stored in ordered B-trees. The make-set operation creates a new tree in O(1) time, the find operation can be performed in O(log n) time to traverse the B-tree from a leaf to the root, and splitting B-trees has been shown to take O(log n) time [13, pages ]. The main open question is the amortized cost of union operations. The union operation may have to merge two sets of items with exactly alternating values, resulting in an O(n)-time operation. On the other hand, the operations take O(log n) to concatenate two trees in the case where all items in one tree are less than all the items in the other tree, as in Ordered Union- Split-Find. Demaine, López-Ortiz, and Munro [7] generalize Mehlhorn s work by solving the problem of merging arbitrarily many sets, and they prove tight upper and lower bounds for the problem. It remains open whether General Union-Split-Find can be solved using operations that take O(log n) amortized time, which would prove the known lower bound tight. In Chapter 3, we present a data structure for which there exists an expensive sequence of operations that forces the data structure to take Θ(log 2 n) amortized per operation; this sequence can be repeated any number of times, so there is a lower bound of Ω(log 2 n) amortized in the worst case. We conjecture that this data structure has polylogarithmic amortized performance. 19

20 1.6 Known Lower and Upper Bounds Table 1.1 summarizes the best known bounds for each of the variations of the Union-Split- Find problem. Both the Cyclic and the General Union-Split-Find problems have not been specifically studied before. Union-Split-Find Lower bound Upper bound Interval Ω(log log n) [14, 18] O(log log n) [14] Cyclic Ω(log w n) [12] O(log w n) (new) Ordered Ω(log n) [17] O(log n) [19] General Ω(log n) [17] O(n) (obvious) Table 1.1: The best known bounds for the four variations of the Union-Split-Find problem. The upper bound for the Cyclic problem applies only to the word RAM model (and thus to the cell-probe model as well). All other bounds apply to all three models of computation under discussion. 20

21 Chapter 2 Cyclic Union-Split-Find Problem Recall that the unrolled linear form of the Cyclic Union-Split-Find problem requires the following operations on a balanced-parenthesis string: insert, delete, and parent. Insertions and deletions of pairs of parentheses must maintain the invariant that the resulting string remains balanced. The pair of parentheses inserted or deleted must also form a matching pair in the context of the entire balanced-parenthesis string. The parent operation returns the immediately enclosing pair of parentheses of a query pair. The language of balanced-parenthesis strings is also classically known as the Dyck language. The problem of simultaneously handling insertions, deletions, and parent queries is one of several natural dynamic problems for the Dyck language. Frandsen et al. [9] discuss the various dynamic membership problems of maintaining balanced-parenthesis strings subject to operations that modify the string and check whether the string is a valid balancedparenthesis string. They establish a lower bound of Ω(log n/ log log n) in the word RAM model for any data structure solving the related problem of supporting insertions, deletions, and a member operation for checking whether the string is in the language. Frandsen et al. also establish a lower bound of Ω(log n/ log log n) in the bit probe model (the cell-probe model wherein the cell sizes are of 1 bit each) for the related problem of supporting the operations of changing a single character in the string and testing the string s membership in the language. Alstrup, Husfeldt, and Rauhe [2] give a data structure that solves the latter of these two problems optimally in Θ(log n/ log log n) in the word RAM model. Their data structure is essentially a balanced tree of degree O(log n) that stores each parenthesis symbol in the leaves of the tree. We use a similar data structure to solve the Cyclic 21

22 Union-Split-Find problem. Theorem 1. Under the standard transdichotomous assumption that the word size is Ω(log n), the Cyclic Union-Split-Find problem can be solved in Θ(log w n) amortized time per operation. We achieve this result by modifying an existing data structure. We store the balancedparenthesis string in the leaves of a strongly weight-balanced B-tree, a data structure first proposed by Arge and Vitter [3] and further improved by Bender, Demaine, and Farach- Colton [4]. The key to achieving the desired time bound is setting the branching factor to be O(w ε ) for some constant ε where 0 < ε < 1. There will therefore be O(log w ε n) = O(log w n) levels in the B-tree. The key to supporting the parent operation quickly in this tree is the following property. Property 1. Given the position i of the query pair s open parenthesis ( in the balancedparenthesis string, finding the open parenthesis of the parent pair is equivalent to finding the last unmatched ( parenthesis in the first i 1 characters of the string. Using this property, we store at each internal node a succinct representation of the unmatched parentheses of the substring stored in the subtree rooted at that node. Given a query pair of parentheses, the parent operation thus walks up the tree from the leaf storing the open parenthesis of the query pair, recursively performing essentially a predecessor query at each internal node, and stopping once evidence of the parent parenthesis has been found. (An equivalent algorithm would be obtained by starting from the close parenthesis and performing successor queries.) We can observe that all queries originating from the children of the parent pair of parentheses stored at the leaves u and v will terminate by the time the algorithm reaches the least common ancestor (LCA) of u and v. This means that nodes above their LCA do not need to know about u or v. For convenience, we will overload terminology and define the LCA of a single node or parenthesis to be the LCA of that node and the leaf that contains the parenthesis it matches in the string. We also define the Parent s LCA (or PLCA) to be the LCA of a query pair s parent pair of parentheses. Note that to support the parent operation at any internal node x, we only need to maintain data for parentheses below it that have their LCA at x or at some ancestor of x. Enough of this data can be compressed into O(1) machine words such that the parent operation only needs to spend O(1) time at 22

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