Patent Application
20080256037
1. Field of the Invention
The present invention generally relates to a system and method for generating an ordered list.
2. Description of Related Art
Online shopping has become an increasingly popular activity and millions of customers use the Web today to purchase items. Customers are usually presented with a fielded search interface using which they can specify selection criteria such as the check-in/check-out dates for a hotel room, the color/model of printer cartridges and the make/model of cell-phones. The items that satisfy the selection criteria are then returned in the order of their price. Travel aggregators and online stores often offer price discounts based on purchasing a certain quantity of items. Such discounts are usually in the form of promotional rules such as “Stay 3 nights, get a 15% discount on double-bed rooms”, “Buy 2 Canon printer cartridges, get the third one free” and “Buy 2 Motorola Razr cell-phones, get $50 off”. Thus, depending on the user query and the properties of an item, only some of these promotional rules may apply. Due to the potentially large number of items and promotional rules, the ability to compute the discounted price for each item at query time and return items ranked by their discounted price, is a key factor in the efficiency of online shopping.
The simplest and most common solution to the above problem is to select the items that satisfy the user query, apply the applicable promotional rules to each selected item, and return the top few items with the lowest price. While this approach performs reasonably well for a small number of items and promotional rules, it suffers from obvious scalability problems when the number of items and promotional rules increases. This problem is particularly bad for travel aggregators such as hotels.com and travelocity.com, which have to issue an expensive web service call to the site responsible for each item to check for its discounted price.
In view of the above, it is apparent that there exists a need for an improved system and method for generating a list of advertisements.
In satisfying the above need, as well as overcoming the drawbacks and other limitations of the related art, a system and method for generating a list of advertisements is provided.
The system includes a query engine and an advertisement engine. The query engine receives a query from the user and determines parameters to match with the advertisement. The advertisement engine receives the parameters and generates a list of items based on the parameters. The system may function in a precompute mode to calculate intervals for each available item to minimize the variable processing costs for each item. For example, the price per unit may vary based on desired quantity. Further, the price per unit may be a function of multiple pricing rules in affect for each item. Accordingly, the pricing rules over a quantity interval may be generalized by the minimum price per unit within the interval. Further, the number of intervals a crossed item may be selected in a manner to satisfy a given space constraint. By characterizing each item by a minimum price within each interval, the system can quickly query the interval matching the desired quantity for each item and determined if the minimum price for that interval is less than the top-k prices already included in the list. If the minimum price is not less than the top-k items on the list, the system can quickly index to the next item. Alternatively, if the minimum prices is less than the top-k price on the list, the item may be added to the list or the actual price may be calculated for further comparison.
Accordingly, when identifying intervals, the system may start analyzing each item using a single interval and continuously increase the number of intervals while determining the split points that yield the maximum processing benefit. As such, the minimum price for each interval is stored along with the processing benefit achieved by adding each interval to an item. Thereafter, the intervals may be combined by optionally smoothing the benefit data and selecting the number of intervals for each item that yields the overall largest processing benefit that can be achieved within the given space constraint.
Further objects, features and advantages of this invention will become readily apparent to persons skilled in the art after a review of the following description, with reference to the drawings and claims that are appended to and form a part of this specification.
Referring now to
Referring again to
The query engine 12 is also in communication with the advertisement engine 16 allowing the query engine 12 to tightly integrate advertisements with the user query and search results. To more effectively select appropriate advertisements that match the user's interest and query intent, the query engine 12 may be configured to further analyze the text query 20 and generate a more sophisticated translated query 30. The query intent may be better categorized by defining a number of domains that model typical search scenarios. Typical scenarios may include looking for a hotel room, searching for a plane flight, shopping for a product, or similar scenarios.
One example may include the text query “New York hotel August 23”. For this example, the query engine 12 may analyze the text query 20 to determine if any of the keywords in the text query 20 match one or more words that are associated with a particular domain. The words that are associated with a particular domain may be referred to as trigger words. Various algorithms may be used to identify the best domain match for a particular set of keywords. For example, certain trigger words may be weighted higher than other trigger words. In addition, if multiple trigger words for a particular domain are included in a text query additional weighting may be given to that domain.
The translated query 30 is provided to the advertisement engine 16. The advertisement engine 16 includes an index module 32 and a data module 34. The advertisement engine 16 performs an ad matching algorithm to identify advertisements that match the user's interest and the query intent. The advertisement engine 16 compares the translated query 30 to information in the index module 32 to determine if each index entry matches to the translated query 30 provided from the query engine 12. The index entries may be ordered in a list from lowest price to highest price for a predefined number of items. The list may be referred to as a top-k list where k represents the predefined number of items. The advertiser system 38 allows advertisers to edit ad text 40, bids 42, listings 44, and rules 46. The ad text 40 may include fields that incorporate, domain, general predicate, domain specific predicate, bid, listing or promotional rule information into the ad data.
The advertisement engine 16 may then generate advertisement search results 36 by ordering the index entries into a list from the lowest priced entries to the highest priced entries. The advertisement engine 16 may then access data entries from the data module 34 that correspond to each index entry in the list from the index module 32. Accordingly, the advertisement engine 16 may generate advertisement results 36 by merging the corresponding data entries with a list of index entries. The advertisement results 36 are then provided to the query engine 12. The advertisement results 36 may be incorporated with the text search results 28 and provided to the user system 18 for display to the user.
A naive way of indexing promotional rules is to precompute and explicitly store the discounted price for each item-quantity pair. Thus, when a user issues a query for a given quantity, the discounted price for the items that satisfy the user query can be looked up directly, and the top few results can be returned to the user. However, this simple approach can lead to a significant space requirement because the number of items and the number of possible quantities can be quite large; this extensive space requirement is particularly undesirable in large online sites, which, store large parts of the data in main-memory to achieve the desired throughput and response time. A related disadvantage of this approach is that the discounted price has to be precomputed for all quantities and items, even though many quantities are rarely queried and many items rarely make it to the top few results.
To address the limitations of the naive approach, a promotional rule associated with an item i is modeled as a function that takes as input a quantity q, and returns the discounted unit price for that quantity. For instance, “Buy at least 2 Motorola cell-phones, get 10% off the unit price” can be modeled as a function f associated with a Motorola cell-phone, where f(q)=p, q=1 and f(q)=0.90×p, q≧2, where p is the regular (non-discounted) price for a cell-phone. This function is illustrated in
However, representing functions as intervals introduces new challenges for query processing: since only store the minimum price for a given item and interval (for space-savings) is stored, some post query processing needs to be done to determine the actual discounted price for each item, and post query processing can be expensive if it has to be done for many intervals. To address this issue, a threshold algorithm can be adapted to prune away a large number of items and intervals that cannot possibly make it to the top few results, thereby greatly reducing the cost of post-processing. A straightforward adaptation of the threshold algorithm would not suffice given that the set of functions that qualify to compute the discounted price of a query answer is only known at query time and varies from item to item. For example, given a query looking for 2 printer cartridges, the rules “Buy 2 Canon printer cartridges of any color, get the third one free” and “Buy at least 2 red printer cartridges of any type, get $5 off the total price” would both apply to a red Canon printer cartridges while only the former one would apply to non-red printer cartridges.
An algorithm is also provided for determining appropriate function intervals for a given set of items and promotional rules. The algorithm takes in a space budget and uses the query workload to identify the items and functions that most need to be split into intervals, and produces a set of intervals that are provably close to optimal. An interesting aspect of the algorithm is that it makes very few assumptions on the nature of functions, and it thus can be applied to a very broad class of promotional rules. Experiments have shown that the proposed approach offers orders of magnitude improvement in performance over other approaches. In particular, it is shown that by increasing the space budget to only 1.5 the size of the database of items, the algorithm is 5 orders of magnitude faster than other approaches.
Items may be stored in the advertisement engine as tuples in a relation, with a distinguished attribute storing the price of the item (without applying any discounts). The notation i.price is used to refer to the pre-discount price of item i. Table 1 shows some items stored in a relation that stores cell-phones.
Similarly, there can be many other relations corresponding to different item categories such as laptops, printer cartridges, etc. Without loss of generality, we will use the Cell-phones relation for examples throughout the instant application.
Promotional rules can be specified at different granularities and can use arbitrary functions to express different discounts. For example, the rule p1 in Table 2 applies to all Motorola cell-phones, while the rule p2 applies to a specific cell-phone model. Finally, the rule p3 applies a fixed discount to the total price of buying Siemens phones only. We capture these semantics by associating a set of promotional rules with each item. For the example shown in Tables 1 and 2, the items with ItemIds 1, 3 and 5 each have exactly one rule associated with them, i.e., p4, p3 and p1, respectively. The item with ItemId 4 has two rules associated with it, p1 and p2, and the item with ItemId 2 has no rules associated with it.
Given an item i and an associated set of rules RSeti, a function can be defined Applyi: RSeti×N→R, which intuitively takes in a rule pεRSeti and a quantity qεN, and returns the unit price for item i for quantity q using only rule p. In our running example, if we denote the Motorola Razr cell-phone as MRC, ApplyMRC(p1, 1)=MRC.price, ApplyMRC(p1, 2)=MRC.price, ApplyMRC(p1, 3)=2×MRC.price/3, and so on. Similarly, ApplyMRC(p2, 1)=MRC.price, ApplyMRC(p2, 2)=0.90×MRC.price, ApplyMRC(p2, 3)=0.90×MRC.price, and so on.
Finally, given item i, RSeti and Applyi, we can define the discounted price function fi: N→R as follows:
f
i(q)=min({i.price}∪∪RεRSct
Intuitively, for a given quantity q, fi(q) returns the minimum unit price for item i obtained by applying a discount rule unless there are no rules applicable to the item in which case the original price of the item is used. Note that there is an implicit assumption in the above definition that only one rule can be applied for an item at a given time. While this assumption is commonly made in many online stores, we can also define fi to allow the application of a combination of rules. For the example of ItemId 4, line 50 in
It will be assumed throughout the remainder of this application that an item I is associated with an arbitrary discounted price function fi. The issue of whether fi is obtained by applying one rule or a combination of rules is immaterial because the subsequent algorithms do not depend on this assumption.
The precompute interval (PI) approach will be considered throughout the remainder of this application. The key idea of this approach is to approximate a function fi by a set of numbers. Specifically, the PI approach splits each fi into one or more quantity intervals, and stores the minimum value of fi for each interval. To see how this helps, consider the rule p4 on Panasonic VS2 phones that was discussed in the previous section. If p4 is split into two intervals, I1 for quantities less than or equal to 2 and I2 for quantities greater than 2, then the minimum prices of f1 for I1 and I2 are good approximations of f1; in fact, the minimum values for I1 and I2 exactly capture f1 in this case and will not incur wasted work. Consequently, the PI approach may avoid wasted work by intelligently splitting fi's into multiple intervals. In order to avoid an extremely large space requirement due to large number of intervals, a space budget (specified as the total number of intervals for all items) is provided as a parameter to the PI approach.
Table 3 shows a possible instantiation of the Intervals table. Each row in the table corresponds to a single interval for a given fi. The first column stores the id of an item i, the second column lowq stores the low range of the interval, the third column highq stores the high range of the interval, the fourth column minfi stores the minimum value of fi for the interval and the final column stores fi. For example, there are 3 intervals associated with ItemId 4; [1, 1], [2, 2], [3, ∞]; each of which is associated with the lowest discounted price value. This is illustrated by the vertical bars in
In the query processing algorithm L is set to be the list of Interval ids that overlap with the query quantity Qty and that correspond to items that satisfy Pred. The computation of L can be optimized using traditional indices such as join indices (for finding the list of Interval ids that correspond to items that satisfy Pred) and interval/segment trees (for finding interval ids that overlap with the query quantity Qty).
Now referring to
With regard to selecting intervals for the PI approach, one key challenge is to use the query workload to determine the best set of intervals that (a) reduce the overall query processing time, to (b) satisfy the space budget constraints. The naive solution to this problem—enumerating all possible sets of intervals—has computational complexity that is exponential in the number of items, which is clearly infeasible. However, some key properties relating fi's and item intervals can be exploited to develop an algorithm that is both efficient and provably close to optimal.
The cost of evaluating a query Q using the PI algorithm (Algorithm 1), can be split into two components of the overall cost. The first component is the fixed cost, which is the cost of evaluating Q, independent of the choice of intervals. The fixed cost has three parts: (1) the index probes (line 1)1, (2) k iterations of the for loop that add the top-k results to the result heap (lines 9-10)2, and (3) the final iteration of the for loop when the termination condition is satisfied (lines 5-6). If we computed and stored all possible intervals, then each query would only incur the fixed cost.
The second component of the cost is the variable cost, which is the cost of evaluating a query after excluding the fixed cost. This component of the cost depends on the choice of intervals. Given a query Q and a specific choice of intervals P, if the Algorithm 1 iterates over its for loop m times, then the variable cost is the cost of evaluating m−k−1 iterations; these iterations correspond to items/intervals that are processed by the algorithm but which never make it to the top-k results. (We arrive at the number m−k−1 because out of the total of m iterations, k iterations are used to produce the actual top-k results, and the last iteration is for the termination condition,)
The total variable cost can be minimized over all queries in a query workload QW=[Q1 . . . ,Qn]. In other words, all cost other than minimum fixed cost that must be incurred for each query Qi can be minimized. Let I be the set of items, and let Ivals be the set of all possible quantity intervals.
Definition 1. Partition. A partition P is a function P:I→2Ivals such that for all iεI, the intervals in P(i) (a) are non-overlapping (to avoid redundancy), and (b) cover the entire quantity range (to avoid missing quantities).
Intuitively, a partition is just a formal way to denote a specific choice of intervals.
Recall that the variable cost of evaluating a query Q using a partition P is defined as the cost of evaluating each one of the m−k−1 iterations (lines 9-10 in Algorithm 1). The cost of each iteration is considered to be a single unit and then define the variable cost of query Q can be defined using partition P, varcost(I,P,Q), to be m−k−1. In addition, the notation culprits(I,P,Q), can be defined which will be used extensively later, to refer to the set of items whose intervals are processed in the m−k−1 iterations of Q that contribute to its variable cost. Therefore, given a set of items 1, the set of all possible quantity intervals Ivals, a query workload QW, and a space budget s, a partition P can be found such that it minimizes the overall variable cost ΣQεQW (varcost(I,P,Q)) subject to the space constraint ΣiεI|P(i)|≦s.
A simple way to identify the partition P is to explicitly enumerate all the partitions that satisfy the space budget, compute the cost for each such partition, and finally pick the partition that has the minimum cost. However, this algorithm is likely to be very inefficient due to the large number of possible partitions. Specifically, if the number of distinct query quantities is t, then the number of possible partitions is ‘2t×|I| s−|I|’. (There are 2t interval split points for each fi, one before and one after every query quantity; thus, the total number of interval split points for all items is 2t×|I|. From these, s−|I| split points may be chosen, since we start with |I| intervals and each additional split increases the number of intervals by one.) Thus, for even modest sized databases, such as one having 10000 items, 10 query quantities and a space budget of 20000, we have ‘2×105 104’ possible partitions!
Fortunately, it turns out that a key property relating partitions can be exploited that dramatically reduces the set of partitions that need to be considered. We first introduce some notation before formally stating the independence property and presenting our algorithm.
Definition 2. Variable Cost of an Item. The variable cost for an item iεI given a partition P and a query workload QW is defined to be:
vc
i(I,P,QW)=|{Q|QεQWiεculprits(I,P,Q)}|
(In this definition, { } refers to a bag, not a set, in order to deal correctly with duplicate queries.)
In other words, the variable cost for an item i may be defined by the number of times the item appears as a culprit in the query workload, i.e., the number of times an interval associated with an item is processed by the PI algorithm without the item being part of the final top-k result. It is easy to see that ΣiεI vci(I,P,QW)=ΣQεQW varcost(I,P,Q), i.e., the sum of the variable costs of all items is the same as the sum of the variable costs of all queries (which in turn is the same as the overall variable cost).
For notational convenience, maxprice(I,Q) is used to denote the maximum price of the top-k results obtained by evaluating Q over I (i.e., the price of the most expensive item in the top-k results). For ease of exposition, we assume that the values produced by evaluating fi's for a given quantity are all unique, although this is not a limitation in practice (for instance, all non-unique fi values can be made unique by appending the id of i).
Lemma 1. Independence Property. Given a set of items I and a space budget s, let AllParts be the set of all partitions that satisfy the space budget. Then, given a query workload QW:
∀iεI,∀P1,P2εAllParts,(P1(i)=P2vc1(I,P1,QW)=vc1(I,P2,QW))
Proof Sketch: Consider a partition PεAllParts and a query Q=(Preds,Qty, k)εQW. Let QtylvalQ,i be the interval in P(i) that contains Qty. (Recall that the P(i)'s are non-overlapping and cover the entire quantity range, so there is exactly one interval that satisfies this condition.) From Algorithm 1, it can be seen that for an item i and query
i is a culprit iff its minimum price in the interval that contains Qty is less than the top-k maximum price. Consequently, vci|{Q|QεQWQ
minqεQtyval
Informally, the property states that the benefit of choosing a particular set of intervals for item i is independent of the choice of intervals for other items. Consequently, the problem can be solved for each item separately, and then combined these to produce the overall solution. The overall complexity of the algorithm that exploits this observation is O(t3×|I|+s log|I|+|I|×|QW|), and it produces a solution that is within a factor (s−|I|−2t+1)/(s−|I|) of optimal (it is shown later that in fact, the complexity of the algorithm is usually much less, especially for the |I|×|QW| component).
The algorithm works in two steps. It first finds the optimal way to choose v intervals, 1≦v≦2t+1, for each item (recall that t is the number of query quantities seen, so there are 2t possible split points, one before and one after each query quantity, and thus a maximum of 2t+1 intervals). It then finds the global optimum by choosing v1, v2, . . . , v|I| such that v1+v2+ . . . +v|I|≦s and choosing vi intervals for item i gives us the globally optimal partition.
As shown in
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Now these steps will be described in more detail. The first step can be solved efficiently using dynamic programming and the second step can be solved using a variant of the knapsack problem.
The current problem is to find for each item i, the optimal way to choose 1 interval, 2 intervals, . . . , 2t+1 intervals. Here, optimal means minimizing the variable cost vci. In order to solve this problem, a Culprits table is created using the query workload. The Culprits table has three columns, ItemId, Quantity and MaxTop−kPrice, and it contains the following set of rows:
((ItemId,Quantity,MaxTop−kPrice)2Culprits|
QεQŴItemIdεculprits(I,P0,Q)
̂Quantity=Q.Qty
̂MaxTop−kPrice=maxprice(I,Q)}
where P0 is the partition in which each item is assigned the one interval that covers its entire quantity range. Intuitively, the Culprits table has one row for each culprit of each query, and the row contains the ItemId of the culprit, the quantity of the query, and the maximum price of the top-k results of the query. Table 4 shows an example Culprits table for different quantity values and queries.
Note that creating the Culprits table does not require additional processing; it can be easily created during regular query processing by initially running the PS approach using the P0 partition, and logging the information for each culprit.
Given the Culprits table, we can determine the value of vci for a given choice of intervals for an item i. As an illustration of how this can be done, consider the item corresponding to ItemId 4 in Table 1, with f4 and intervals shown in
Recall that the value of vci should be minimized for a given number of intervals v. Thus, in pictorial terms, v intervals should be chosen such that the number of points above the MVLs is minimized. Since it is convenient to think of this problem as a maximization problem, we can equivalently view the problem as maximizing the number of points below the MVLs. Thus, the benefit can be defined for each interval to simply be the number of points below its MVL, and then a set of intervals can be found such that the total benefit is maximized. More formally, for interval Ival of item i, its benefit can be defined as:
B
T(Ival)=|{(ItemId,Quantity,MaxTop−kPrice)εCulprits|ItemId−i.idMaxTop−kPrice<minqεIvalfi(Quantity)}
and the best benefit for item I is broken into v intervals:
Given the above definitions, a dynamic programming algorithm can be used to find the total benefit for the optimal set of intervals.
Algorithm 2 shows the pseudocode. The algorithm is similar to the dynamic programming algorithm for finding the VOPT histogram, which also finds optimal intervals of a query range but for a different context (query result size errors, as opposed to culprits in our case).
The algorithm is run on each item. The initialization phase first computes the benefit for every interval. Then, for each point between 1 and 2t+1, the algorithm computes the best number of intervals generated up to that point. The best number of intervals is computed in line 5 as the maximum benefit of a choice of intervals for that point. The naive implementation of the algorithm, run for all items, takes time Q(t3×|Table|), where |Table| is the size of the Culprits table; the t3 comes from the for-loops of the algorithm and |Table| comes from repeated calls to the Benefiti(Ival) function, which can access all rows associated with an item for each call.
A key observation regarding the Culprits table is that its rows can be aggregated to record the number of culprits instead of each culprit individually. In this case, the cumulative benefit for each interval can be pre-computed in the initialization phase. This makes the running time of the algorithm essentially independent of the size of the Culprits table. The complexity is thus reduced to O(t3×|I|+|Table|), which is usually much smaller than Q(t3×|Table|).
In the previous subsection, how to break the interval of a given item into v pieces was described in such a way that the number of avoided culprits was maximized, for any given v. For the i-th item, we denoted this number by BestBenefiti(v). Recalling that a storage constraint limits the use of most s items, we v1+v2+ . . . +v|I|≦s is found such that BestBenefit1(v1)+ . . . +BestBenefit|I|(v|I|) is as large as possible.
Throughout, it is assumed that each item will be broken into at most 2t+1 pieces. For each i and j, the incremental improvement is tracked of using j+1 intervals to describe the i-th item, instead of just j. cij is used to denote that improvement.
c
ij=BestBenefiti(j+1)−BestBenefiti(j).
Notice that Σj=ik cij=BestBenefiti(k+1) since the sum telescopes. Thus rephrase our problem as finding k1+ . . . +k|I|≦sdiff such that Σi=1|I| Σj=1ki Cij is maximized. (For readability, sdiff=s−|I| is defined throughout this section.)
As a running example, Table 5 contains several items and their interval benefits. The item with ItemId 4, for example, contains the sequence 0, 7, 2, indicating that using two intervals gives no benefit over using one, while using three intervals gives a benefit of 7 over using two intervals, and using four intervals gives a benefit of 2 over using three intervals. (That is, c41=0, c42=7, C43=2.) For simplicity in our example, we assume that there are only four items in I.
There is a dynamic programming algorithm to solve this problem exactly. Continuing the above example with sdiff=5, this algorithm would take 5, 4 from the item with ItemId=6; it would take the 8, 4 from item 7; and it would take the 4 from item 8. Thus, the total benefit is 25, and the algorithm indicates that item 4 should be described with just one interval, items 6 and 7 using three intervals, and item 8 using 2 intervals.
Although the dynamic programming algorithm works in polynomial time, the approach takes O(sdiff×|I|) time just to execute its outer loop. Since sdiff and |I| are both extremely large, this approach is impractical, even in our off-line setting.
However, we note that if cij≧cij for all i and all j<j′, the exact solution can be found very efficiently using the greedy algorithm: Simply find the sdiff largest cij, where if cij=cij′ with j<j′, then the tie us broken in favor of cij. For each i, let ki be the largest index such that the algorithm took ciki. Since cij≧cij′ for all j≦j′, it is not hard to see that the algorithm must have taken ci1, ci2, . . . , ciki. Hence, k1+ . . . +kN=sdiff, and we have the optimal sum since we have the largest sdiff values. For example, if we ignore the item with ItemId=4 in Table 5, then we have cij≧cij for all i and all j<j′. Thus, if sdiff=5, we can simply pick the largest sdiff values, which correspond to 5, 4 for item 6, 8, 4 for item 7 and (the first) 3 for item Note that finding the top sdiff values from |I| lists can be done extremely efficiently. By maintaining a pointer into each list and having a heap-like structure, we can find the top sdiff values in O((sdiff+|I|)log|I|)=O(s log |I|) time.
Unfortunately, cijs will not be decreasing in general. In fact, Table 5 produced from
So in general, it is not the case that cij≧cij′ for all I and j<j′. However, it is still possible to efficiently find a provably good approximation to the optimal solution. The approach is to “smooth” the cij to produce c′ij such that c′ij 2c′ij′ for all i and j<j′, along with other properties. Using this technique, a solution may be found at least (sdiff−t)/sdiff times as good as optimal. Since we expect sdiff is expected to be thousands of times larger than t in practice, this shows that the approximate solution is better than 99.9% of optimal.
As an illustration of the smoothing technique, consider again the item with ItemId 4 in Table 5. Intuitively, the 7 is preferred. However, the 0 is used first. So the 0, 7 may be replaced with two copies of their average: 3.5, 3.5. Notice that taking 0, then 7, is helpful exactly when taking 3.5 followed by 3.5 is helpful. Continuing, the 2, 4 are replaced with two copies of their average: 3, 3. In general, the prefix sequence is found with the largest average; this may simply be the first item of the sequence. Then each of those values is replaced with the average, and recursively iterated on the remaining sequence. Since items 6, 7, and 8 already have cij that are decreasing, nothing is done for those items. The smoothed values are provided in Table 6.
With the smoothed values c′ ij in hand, we simply find the sdiff largest values, where if c′ij=c′i′j′, then we break ties in favor of c′ij if i<i′; if i=i′ as well, we break ties in favor of c′ij when j<j′. As we noted above, this can be done in O(s log |I|) time.
To illustrate, consider the example, now with sdiff=8. the heap is initialized with the values 3.5, 5, 8, 3 (taking O(|I| log |I|) time), and a pointer is maintained to the first element in each item's list. The maximum value is extracted from the heap, 8, in O(lg |I|) time, and update the pointer for item 7 to point to the second element in its list. Then this value (in this case, 4) is added to the heap. Repeating this, the maximum value, now 5, is extracted and the pointer for item 6 is updated to point to the second item in its list. This value, 4, is added to the heap. On the third iteration, 4 is extracted and 1 (the third item in the list for item 6) is inserted. Then 4, 4, 3.5, 3.5, and 3 are extracted. Hence, the smoothed values that were extracted include 8, 5, 4, 4, 4, 3.5, 3.5, 3 corresponding to the original values 8, 5, 4, 4, 4, 0, 7, 2. Notice that the sum of the smoothed values 3.5+3.5 are exactly equal the original values 0+7. However, the last smoothed value that was extracted, 3, corresponds to 2. In general, at most the last 2t+1 values (which all come from the same item) will be overestimates of the original values. Thus, when translating the c′ij back to the original cij, the total benefit obtained using these smoothed value is at least (sdiff−2t+1)/sdiff of optimal.
For the sake of completeness, an outline of a smoothing algorithm is provided. For readability, the notation
Essentially, the algorithm starts at a cij and looks ahead to see if there is any subsequent cij′ that can increase the average value of all intermediate cik, j≦k<j′. As can be seen, this algorithm has complexity O(t2).
The overall complexity of finding a nearly optimal partition is the sum of the complexity of processing the query workload, plus the complexity of generating intervals for individual items, plus the complexity of finding the optimal combination of intervals across items. As was already noted, processing the query workload takes at most O(|I|×|QW|) time, although this is actually the size of the log, which will usually be much smaller. The running time to find optimal partitions for each item takes a total of O(t3×|I|) over all items. (ignoring the cost of processing the Culprits table, since it is subsumed in the processing time of the query workload.) The running time for finding a nearly optimal combination of intervals across times is O(s log |I|), and smoothing takes O(t2×|I|). Hence, the total complexity is O(t3×|I|+s log |I|+|I|×|QW|).
Novel techniques are presented to evaluate top-k queries over data items whose score is dynamically computed using functions. The functions may be promotional rules which apply to different item quantities. The techniques applied rely on pre-computing appropriate quantity intervals per item and use them to prune items that do not make it to the top-k result. Experiments show that query evaluation using quantity intervals is scalable in the number of items and functions and performs several orders of magnitude better than the naive approach.
Although the above examples relate to shopping for a cell phone, the algorithm is also applicable to shopping for hotel rooms or entirely different applications such as searching traffic routes. As such, an on-line map may rank routes by predicting a congestion level, where the congestion score is a function of the time of day being queried. Accordingly, the quantity of items purchased, from the shopping example, corresponds to the time of day. As such, the congestion score is a query dependent scoring relationship. Destination and origin addresses may be used to find a list of the top-k least congested routes between two addresses. The congestion for a particular time of day may be estimated by rules such as “at 3:00 p.m., congestion level on Highway 280 in a ten mile radius around Palo Alto is high.” Further, the rules may even be inferred from past traffic data. Similar to the price of cell phones, the congestion level is not constant but is a function of the time of day and can be characterized by intervals.
In alternative embodiments, dedicated hardware implementations, such as application specific integrated circuits, programmable logic arrays and other hardware devices, can be constructed to implement one or more of the methods described herein. Applications that may include the apparatus and systems of various embodiments can broadly include a variety of electronic and computer systems. One or more embodiments described herein may implement functions using two or more specific interconnected hardware modules or devices with related control and data signals that can be communicated between and through the modules, or as portions of an application-specific integrated circuit. Accordingly, the present system encompasses software, firmware, and hardware implementations.
In accordance with various embodiments of the present disclosure, the methods described herein may be implemented by software programs executable by a computer system. Further, in an exemplary, non-limited embodiment, implementations can include distributed processing, component/object distributed processing, and parallel processing. Alternatively, virtual computer system processing can be constructed to implement one or more of the methods or functionality as described herein.
Further the methods described herein may be embodied in a computer-readable medium. The term “computer-readable medium” includes a single medium or multiple media, such as a centralized or distributed database, and/or associated caches and servers that store one or more sets of instructions. The term “computer-readable medium” shall also include any medium that is capable of storing, encoding or carrying a set of instructions for execution by a processor or that cause a computer system to perform any one or more of the methods or operations disclosed herein.
As a person skilled in the art will readily appreciate, the above description is meant as an illustration of the principles of this invention. This description is not intended to limit the scope or application of this invention in that the invention is susceptible to modification, variation and change, without departing from spirit of this invention, as defined in the following claims.
