Patent Application
20110264482
Resource matching is a common problem involving allocation of limited resources in an efficient manner. The resource matching problem can present itself in numerous ways, and involve a wide variety of limiting criteria to be considered in finding solutions. Some examples of resource matching problems include matching people to tasks, and matching physical resources to jobs utilizing the resources, such as assigning aircrafts to scheduled flights, allocating bandwidth to radio stations, and scheduling computing devices to run software applications. Other problems that can benefit from resource matching include optimizing register allocation during computer program compiling (e.g., such that the most frequently used values of the compiled program are stored in the fast processor registers), and pattern matching applications (e.g., coding and un-coding messages)
Some of the attributes of high quality resource matching include fast speed, maximized matching, respecting limiting criteria, accommodation of established priorities, flexibility, and consideration of secondary objectives. Various previous approaches have been used to resolve resource matching problems, including Integer Linear Programming approaches, bipartite graph matching, and brute force trial and error methodologies. However, some methodologies are not sufficiently time-efficient for use in real-time applications. Some previous approaches can produce good quality (matching) results in acceptably short periods of time, but can become trapped in local optima, and/or can produce worst-case scenarios that are not useful. Some are adept at finding maximum matching, but are too complex and/or too slow to be deployed in real time against complex problems that are also dynamic in nature.
Methods, computer-readable media, and systems are provided for resource matching. One computer-implemented method for resource matching includes logically arranging a plurality of relationships between a number of tasks and a quantity of resources for performing tasks into a multi-dimensional augmenting tree order. The relationships correspond to edges of a multi-dimensional bipartite graph matching solution between nodes representing tasks and resources respectively. First, an un-matched task from the multi-dimensional bipartite graph matching solution is established as a root node. Additional edges and nodes are added beginning from the root node with an un-matched edge and alternating between matched and un-matched edges until a color-unmatched resource is a leaf node. The multi-dimensional augmenting tree order is modified by un-matching matched edges and matching un-matched edges. Finally, a quantity of matched edges for the modified multi-dimensional augmenting tree order is determined.
Individual jobs 102 may be performed using a suitable resource (not reflected in
Jobs 102 are shown in
One resource matching problem of interest occurs in the allocation of shared computing resources across an organization (e.g., enterprise), such as might occur for software application testing and collaboration to achieve more performance testing in less time. For example, a large and/or complex organization may have an enterprise-wide testing platform. A centralized control is established to coordinate use of computing resources deployed across the entire enterprise such that they may be shared, and thus utilized more efficiently for the organization as a whole. However, such sharing creates a problem of coordination, and requires appropriate scheduling of computing resources, some of which may be suitable for running particular jobs, and not suitable for running others.
The coordination problem can be very complex, demanding the efficient allocation of computing resources for possibly hundreds of thousands of jobs on thousands of computing resources across months of real time, for example. Depending on the sets of capabilities and constraints involved, such scheduling problems of this sort can be very difficult to solve, being NP-complete. NP-complete problems are a class of problems that are so difficult that even the best solutions cannot consistently determine their solutions in an efficient way. In theoretical terms, NP-complete problems refer to those problems that are solved in polynomial time, such as by using a computer capable of making guesses and checking them in polynomial time (e.g., a nondeterministic Turing machine).
One example enterprise-wide testing platform can provide users an ability to configure, edit, and run tests (e.g., of software applications) using an available pool of physical computing resources, each of which can be used by at most one job at a time. A user desiring to run a particular software application test might choose the type and quantity of computing resources required for the particular test. The user may also have to specify other parameters involved in the scheduling problem, such as the estimated amount of time that is forecast to be needed for the testing involved (e.g., job). Users may specify whether a job is to start immediately (e.g., as soon as possible), or whether they wish to reserve computing resources for a particular specified later time (e.g., make a reservation for access to computing resources in the future).
Different users may have different priority access to the one or more of the computing resources (e.g., computing resources may be dedicated to a particular portion of an organization or procured in support of a particular business purpose, etc.), and/or may have different levels of control over certain scheduling attributes. For example, certain users may have administrator privileges, such that they can modify the pool of resources available, or certain particular portions of the pool of computing resources. Thus, a scheduling solution may have to accommodate a complex set of priorities and limiting conditions.
The scheduling problem presented is dynamic, as new jobs need to be scheduled, jobs are completed, resources are added or removed from the pool, and/or priorities of users or jobs shift. Some scheduling issues and/or resource management issues, such as resource pool modifications, may require reallocation, or possibly cancellation of existing computing resource reservations. Therefore, a resource matching solution must efficiently address the real-world constraints and complications.
The resource matching methodology and system of the present disclosure is relatively fast in comparison to previous approaches, typically yielding a solution to a particular complex resource matching problem within a few seconds depending on the complexity of the resource matching problem being addressed. Therefore, it can be deployed on-line in real time. It can accommodate multiple users accessing and using the system simultaneously since re-allocation of resources due to a change in jobs and/or resources is compartmentalized into a smallest possible time period. That is, re-allocation is time-limited such that only a portion of the entire scheduling problem need be re-computed.
Furthermore, the resource matching methodology and system of the present disclosure is efficient in that it maximizes the quantity of job matches. While it may not always be possible to find a complete resource matching solution in all cases given the complex nature of some scheduling problems with respect to a given amount of computing resources, maximizing resource matching facilitates the highest possible efficiency in utilizing available computing resources.
While the resource matching methodology and system of the present disclosure is described in the context of matching software application jobs to computing resources suitable for running particular jobs, embodiments of the present disclosure are not limited to the example application, and may be applied to other types of resource matching problems, and implemented by, and as part of, other systems.
Referring again to
Time starts at time 0 and is divided into time periods 104. Time period 1 is defined to include the time from time 0 to time 1. In some embodiments, time periods 104 can be defined by a particular interval of time (e.g., a shortest duration of time during which it is possible to schedule a job 102). In various embodiments, time periods 104 can be defined as lengths of time equal to the length of time required to complete a job 102 having a shortest duration. For example, in
A time step is the largest period of consecutive time period where all jobs overlap. Jobs need not overlap for the entire duration of the time step, rather each job within a particular time step must overlap each other job(s) within the particular time step. Furthermore, each time step includes at least one completed job. The reader will appreciate that a job that starts at or after the completion of a prior job does not overlap the prior job, and thus, cannot be located within the same time step as the completed prior job. The only exception is the first job, since there are no jobs that supersede (e.g., are completed before) the first job.
Time steps are non-overlapping intervals of time. Jobs may traverse time steps (e.g., cross a boundary between two time steps). However, every job that traverses two time steps 106 overlaps with all other jobs within each of the two traversed time steps 106. In one or more embodiments, the beginning of a time step 106 can be defined as either the beginning of the first job, or the ending of the previous time step. Due to the job overlap constraints of a time step discussed above, time step boundaries usually occur at the beginning or end of a job.
For example, referring to
As shown in
The reader will appreciate that time steps are defined by the particular durations and arrangement of the jobs to be accomplished. Therefore, as the arrangement and/or length of jobs change, the arrangement of time steps (and time closures described below) can change.
As can be seen from
Time closures 108 are defined as a set of consecutive time steps 106 where each two adjacent time steps within a particular time closure have at least one job in common (e.g., at least one job occurs during adjacent time steps). Time closures have boundaries at time steps that have no overlapping jobs. In
According to one or more embodiments of the present disclosure, resource matching is independent between closures. That is, if a new job arrives, or an existing job is removed, only the closure to which that job belongs (assuming it resides completely within one closure) needs to undergo revised resource matching (e.g., re-allocation of resources).
The resource matching problem illustrated in
The resource matching problem between jobs, suitable resources, and time steps, as defined by task scheduling diagram 200 and the suitable resource availability summarized in Table 1 can be set forth on a multi-dimensional (e.g., colored) bipartite graph.
Graph coloring is a special case of graph labeling. Graph coloring is an assignment of labels, traditionally called “colors” to elements of a graph subject to certain constraints. Coloring codes elements of a graph such that no two adjacent like elements share a same color. In visually-displayed graphs, color can be used to color the components of a graph. In computer-implemented graphing solution, coloring can be accomplished by tracking the constraints (e.g., attributes) using codes, flags, variables, or other such information.
Colored bipartite graphs are so-named for using color to convey one or more additional dimensions of information, beyond that which can be conveyed by a two-dimensional black and white bipartite graph. Although the examples and descriptions are presented herein in black and white, embodiments of the present disclosure are not so limited. For example, the solid and dashed lines used herein to designate particular time steps could be implemented as solid lines of differing colors (e.g., a colored bipartite graph). Of course, the presentations of graphical bipartite graphs are intended to facilitate human understanding, and one having ordinary skill in the art will recognize that the methods of the present disclosure may be computer-implemented with, or without, the graphical diagrams being output. That is, a computing device can calculate and track the information set forth in the multi-dimensional bipartite graph using a corresponding quantity of variables, flags, and other settings.
Accordingly, the conventional term of colored bipartite graphs are referred to herein as multi-dimensional bipartite graphs since use of color in calculating, arranging, and/or displaying the information contained in a colored bipartite graph can be implemented without the use of color. The discussion above also applies herein to augmenting tree diagrams. That is, augmenting tree diagrams that typically convey some information using color are referred to herein as multi-dimensional augmenting tree diagram since embodiments of the augmenting tree diagrams of the present disclosure are not limited to augmenting tree diagrams having color. Certain information may be conveyed using pattern (e.g., solid, dashed), or accounted for in computer-implemented methods by variables, flags, or other settings.
The job scheduling problem described by the job schedule shown in
Edges (e.g., lines representing relationships between vertices) of the multi-dimensional bipartite graph 301 represent resource requirements and time steps are represented by colors over the edges. Edges can have more than one color. As the multi-dimensional bipartite graph 301 is depicted here in black and white, rather than color, a first “color” is represented by solid lines (e.g., 314) and a second “color” is represented by dashed lines (e.g., 316). The events of a particular time step are represented by a corresponding “color.” In
Each edge is further labeled with the time step within which the corresponding job is contained. For example, edge 314 is shown with a solid “colored” line, further designated with a time step label 318 of 0, since job J1 occurs in time step T0. Edge 316 is shown with a dashed “colored” line, and further designated with a time step label (e.g., 320) of 1, indicating that job J2 occurs in time step T1. Edge 319 is shown with both a solid and dashed “colored” lines, and further designated with a time step label 321 of “0, 1” to indicate that job J3 occurs in both time steps T0 and T1 (e.g., spans time steps T0 and T1).
In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided unto two disjoint sets such that every edge connects a vertex in one set (e.g., left side of the bipartite graph) to a vertex in a second set (e.g., right side of the bipartite graph). That is, the two sets are independent.
Bipartite graphs are useful for modeling matching problems. In computer science applications, one mathematical modeling tool used in analysis and simulations of concurrent systems is a bipartite directed graph with two sets of nodes: a set of “place” nodes that represent resources, and a set of “event” nodes that generate and/or consume resources. This arrangement is sometimes referred to as a Petri net.
In the case illustrated in
One resource matching limitation can be that not all resources are necessarily suitable for performing all jobs. The limitations for the example scheduling problem depicted in
The thin (e.g., not thick) edges shown in
The reader will understand that edge thickness, graphically indicating matching on a multi-dimensional bipartite graph, can be accounted for in a computer-implemented method by variable value, flags, or other settings. That is, the methods illustrated by the diagrams herein are not limited to graphical embodiments, and may be mathematically and/or logically implemented without a graphical output.
The possible resource matching solution illustrated in
As one skilled in the art will appreciate, solving the resource matching problem using a bipartite graph methodology consists of matching jobs and resources such that there is no resource with more than one matched edge sharing a color (e.g., no two jobs matched to the same resource at the same time). In the possible solution depicted on multi-dimensional bipartite graph 401, resource R1 does not have more than one matched edge (e.g., thicker line) sharing a color (e.g., solid or dashed). The possible solution illustrated in
According to one or more embodiments of the present disclosure, high cardinality matching (e.g., a large proportion of matches) in multi-dimensional graphs can be found by searching for multi-dimensional augmenting trees derived from sub-optimal multi-dimensional bipartite graphs depicting possible solutions to the particular resource matching problem. A sub-optimal solution is one in which all jobs are not completed as scheduled, and there exists at least one augmenting tree rooted in one of the uncompleted jobs.
Construction, as used herein, is not limited to the graphically-depicted assembly of a multi-dimensional augmenting tree diagram (e.g., 530) from a multi-dimensional bipartite graph (e.g., 401). That is, actually producing the multi-dimensional augmenting tree diagram from the multi-dimensional bipartite graph is not required. The diagrams are shown in describing the methods of the present disclosure for ease of understanding. Rather, construction is understood to mean logically arranging the relationships between the jobs and resources as can also be depicted by a multi-dimensional bipartite graph solution and/or a multi-dimensional augmenting tree diagram ordering the relationships according to an order as might be shown as branches of a tree. The reader will appreciate that a computer-implemented method can determine and relate the relationships represented in the various diagrams without having to actually produce a graphical image of the respective diagrams.
Continuing on with diagram 530 shown in
According to one or more embodiments of the present disclosure, the un-matched edges of the augmenting tree diagram are matched, and the matched edges are unmatched to arrive at modified augmenting tree diagram. For the example illustrated in
The reader will appreciate that in the above-described and illustrated case, by matching the un-matched edges, and un-matching the matched edges results in a modified multi-dimensional augmenting tree diagram, the quantity of matches in the modified multi-dimensional augmenting tree diagram has increased by one, thereby providing a better possible solution that depicted by multi-dimensional bipartite graph 401.
Diagram 700 also graphically depicts a second set 770 of possible multi-dimensional bipartite graph matching solutions, 772, 774, and 776, that result in two jobs being completed. For example, solution 772 may represent job J1 being completed in time step T0 by resource 1 and job J2 being completed in time step T1 by resource 1. The multi-dimensional bipartite graph corresponding to this possible solution is shown in
Finally, diagram 700 also graphically depicts a third set 778 encompassing a single possible multi-dimensional bipartite graph matching solutions, 780, that results in three jobs being completed. For example, solution 780 may represent job J3 being complete in time steps T0 and T1 by resource 1 and job J1 being completed in time step T0 by resource 2 and job J2 being completed in time step T1 by resource 2. Since all jobs are completed, the third set 778 of solutions is not sub-optimal (e.g., is an optimal solution), and provides a most efficient use of the available resources.
The method of the present disclosure is not a brute force method for arriving directly at the most optimal solution. Rather, the method of the present disclosure attempts to increment the quantity of matches by one with each iteration, starting with a possible multi-dimensional bipartite graph matching solution, finding an augmenting tree, matching un-matched edges of the tree and un-matching matched edges of the tree to arrive at another matching which increase by one the quantity of matched jobs. The reader will appreciate that by definition, an augmenting tree is a tree that augments by one the quantity of matching with respect to the multi-dimensional bipartite graph used to derive the tree. While a multi-dimensional tree diagram can always be derived from a multi-dimensional bipartite graph, the derived multi-dimensional tree diagram may, or may not, be a multi-dimensional augmenting tree. Thus, whenever a multi-dimensional augmenting tree can be found, the existing matching (e.g., from the multi-dimensional bipartite graph used to derive the multi-dimensional augmenting tree) can be augmented by one.
Entering the range of possible resource matching solutions shown in diagram 760 at one of perhaps many possible solutions is indicated by arrows 782 and 784 in
The reader will appreciate that the method of the present disclosure could be initiated by entering the range of possible solutions at a different particular solution, as may be indicated by arrow 788 in
While a modified multi-dimensional augmenting tree diagram was found for the simple example resource matching problem illustrated and described above, in some instances a modified multi-dimensional augmenting tree diagram (e.g., that increases the size of multi-dimensional matching by one) may not be found for a particular un-matched job. According to one or more embodiments of the present disclosure, an attempt is made to find a modified multi-dimensional augmenting tree diagram for each un-matched job. According to some embodiments, the process can be repeated for other sub-optimal (e.g., less than optimal) solutions as well.
Where several solutions are found having equivalent quantities of matches, other criteria can be utilized to select among the several solutions to determine a best solution (e.g., includes maximum matching and satisfies one or more additional criteria). For example, possible solutions 772, 774, and 776 each represent two jobs being completed. However, the solutions that include completion of job J3 and one of jobs J1 or J2 may yield more efficient use of the resources than a solution that completes jobs J1 and J2, but not the longer job J3.
While the embodiments or the present disclosure have been described and illustrated in graphical form in this disclosure, one having ordinary skill in the art will recognize that the methods thus described are not limited to graphical representations and/or methodologies. That is, the one having ordinary skill in the art will understand how to implement the concepts illustrated graphically herein, using computing devices to find computational solutions without graphical representations or user intervention. The graphical illustrations provided herein are for ease of understanding, and for compact and simple presentation, for benefit of the reader, and are not intended to limit the embodiments of the present disclosure.
Embodiments of the present disclosure can be implemented by one or more computing devices, with or without the graphical output of the diagrams disclosed herein. For example, a resource matching system can include a non-transitory computer-readable medium communicatively coupled to the one or more computing devices. The non-transitory computer-readable medium can include instructions executable on the one or more computing devices (e.g., processors) to perform the methods described herein. The results of the methods of the present disclosure can be output to a printing device, and/or display device, and/or otherwise used to control the actual performance of particular jobs by particular resources.
Embodiments of the present disclosure can be computer-implemented, for example, using the underlying relationships. A matching is a set of color disjoint edges. An edge can be represented as (Job,Resource,Color). Two edges (J1,R1,(C1, . . . , Cn)) and (J2,R2,(K1, . . . , Km)) are disjoint if and only if:
It should be appreciated that the number of disjoint edges is the size of the matching. The goal of the methods of the present disclosure is to find a matching with the maximum size (e.g., the greatest number of disjoint edges). Given a multi-dimensional bipartite graph, and a matching M of size n, if an augmenting tree T (as defined previously herein) can be found from the multi-dimensional bipartite graph, then the matching M (e.g., of the multi-dimensional bipartite graph) can be augmented by 1 to yield a matching M1 with a size n+1. As was described previously, and illustrated in
The above-described methodology can be implemented iteratively. For example, an existing valid matching, or an empty matching, can be used as a starting point from which to begin the iterative process of searching for an augmenting tree from multi-dimensional bipartite graph. Where an augmenting tree is found, a corresponding multi-dimensional bipartite graph can be used to search for another augmenting tree. The iterative process continues until no augmenting tree can be found from a particular multi-dimensional bipartite graph (e.g., all un-matched nodes of the multi-dimensional bipartite graph are used as a root node from which a tree ordering is constructed).
While the above-described iterative process can be started from any existing valid matching, or empty matching, the reader might appreciate that if the jobs are strictly ordered (e.g., by some priority), and if the process is begun from an empty matching using a highest priority job and searching toward progressively lower priority jobs, then the above-described methodology produces a maximum matching (e.g., a matching with the maximum size possible). In this way, optimal solutions can be arrived at more efficiently if some additional information is utilized (e.g., job priority) in deciding upon a starting point for the iterative process described above.
Although specific embodiments have been illustrated and described herein, those of ordinary skill in the art will appreciate that an arrangement calculated to achieve the same results can be substituted for the specific embodiments shown. This disclosure is intended to cover adaptations or variations of one or more embodiments of the present disclosure. It is to be understood that the above description has been made in an illustrative fashion, and not a restrictive one. Combination of the above embodiments, and other embodiments not specifically described herein will be apparent to those of skill in the art upon reviewing the above description. The scope of the one or more embodiments of the present disclosure includes other applications in which the above structures and methods are used. Therefore, the scope of one or more embodiments of the present disclosure should be determined with reference to the appended claims, along with the full range of equivalents to which such claims are entitled.
In the foregoing Detailed Description, some features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the disclosed embodiments of the present disclosure have to use more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed embodiment. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separate embodiment.
