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Certification of Computational Results Greg Bronevetsky

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Background Technique proposed by Gregory F. Sullivan Dwight S. Wilson Gerald B. Masson All from Johns Hopkins CS Department.

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Overview Trying to do fault detection without the severe overhead of replication. Certification Trails are a manual approach that has that programmer provide additional code to have the program check itself. A program generates a certification trail that details its work. A checker program can use this trail to verify that the output is correct in asymptotically less time. Several examples provided. No automation.

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Roadmap We will cover some algorithms to which the Certification Trails technique has been applied Sorting Convex Hull Heap Data Structures The addition of Certification Trails and the creation of the Checker is done manually by the programmer in all cases.

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Trail for Sorting In order to verify the output of a sorting algorithm we must check that The sorted items are a permutation of the original input items. The sorted items appear in a non-decreasing order in the sorter's output. Thus, the trail should contain all the items in their original order, each labeled with its location in the sorted list.

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Sorting Checker A Sorting Checker must: Use the labels to place all elements into their sorted spots and verify that this results in a non- decreasing order. Verify that no two elements are placed in the same location in the ordered list. The Sorter takes O(n 2 ) or O(n log n) time. The Checker takes O(n) time. Checker is asymptotically faster than Sorter.

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Convex Hull Problem Given a set of points on a 2D plane, find a subset of points that forms a convex hull around all the points.

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Convex Hull: Step 1 P 1 is the point with the least x- coordinate. P6P6 P2P2 P8P8 P3P3 P5P5 P1P1 P7P7 P4P4 Points sorted in order of increasing slope relative to P 1

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Convex Hull: Invariant P6P6 P2P2 P8P8 P3P3 P5P5 P1P1 P7P7 P4P4 All the points not on the Hull are inside a triangle formed by P 1 and two successive points on the Hull.

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Convex Hull: Invariant P6P6 P2P2 P8P8 P3P3 P5P5 P1P1 P7P7 P4P4 We know that P 3 is not a Hull point because the clockwise angle between lines and is ≥ 180º. ≥ 180º

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Convex Hull: Invariant P6P6 P2P2 P8P8 P3P3 P5P5 P1P1 P7P7 P4P4 < 180º Note that if clockwise angle between lines and is < 180º, then P 3 is a Hull point

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Convex Hull Algorithm Add P 1, P 2 and P 3 to the Hull. (Note: P 1, P 2 and P n must be on the Hull.) For P k = P 4 to P n... trying to add P k to the Hull... Let Q A and Q B be the two points most recently added to the Hull: While the angle formed by Q A, Q B and P k ≥180 remove Q B from the Hull since it is inside the triangle: P 1, Q A, P k. Add P k to the Hull.

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Trail for Convex Hull Augment Program to Output {q 1, q 2,..., q m } = the indexes of the points on the hull. Output a proof of correctness for {x 1, x 2,..., x r } = all points not on the Hull in the form of the triangle that contains it.

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Convex Hull Checker Checker must check that: There is a 1-1 correspondence between input points and {q 1, q 2,..., q m } U {x 1, x 2,..., x r }. All points in the triangle proofs correspond to input points. Each point in in the triangle proofs actually lies in the given triangle. Every triple of supposed Hull points forms a convex angle. There is a unique locally maximal point on the hull.

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Asymptotic Runtimes Original Convex Hull Algorithm takes O(n log n) time to sort and the Hull construction loop takes only O(n) time. O(n log n)-time total. Convex Hull Checker runs thru the set of points once for each check. O(n)-time total. Checker asymptotically faster than Original.

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Certification Trails for Data Structures Lets have a data structure for storing value/key pairs, ordered lexicographically: (key, val) < (key', val') iff val

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Data Structure Specs Data Structure Operations changekey(key, newval): executed when the pair (key, oldval) exists in the data structure. Removes this pair and inserts the pair (key, newval) deletemin(): deletes the smallest pair (according to the ordering). Returns “empty” if the data structure contains no pairs. predecessor(key): returns the key of the pair that immediately precedes key's pair or “smallest” if there is no such pair. empty(): returns whether the data structure is empty.

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Data Structure Implementation Such a Data Structure can be implemented via an AVL tree, a red-black tree or a b-tree. Most operations will take O(log n) time. We can augement implementations to generate a certification trail: insert(key, val): output the key of the predecessor of the newly inserted pair (key, val). If there is no predecessor, output “smallest”. changekey(key, newval): output predecessor of the new pair (key, newval). If there is no predecessor, output “smallest”.

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Data Structure Checker A Checker for any program using the above data structure can use the certification trail to implement a much faster data structure. All operations can be done in O(1) time. Resulting program will be faster than original program. Maybe asymptotically faster.

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Optimized Data Structure A doubly linked list of (key, val) pairs, sorted according to the pair ordering relation. An array indexed by keys, containing pointers to (key, val) pairs corresponding to the indexes. The first pair (with key=0) contains value=sm, which is defined to be smaller than any other possible value.

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Optimized Data Structure Optimized data structure operations: insert(key, val): Read from trail prec_key = the key of the pair preceding the new (key, val) pair. Check that it is a valid index. Look at the pair pointed to by array[prec_key]. Verify that it is ≠null. Place the (key, val) pair at index key, following the (prec_key, prec_val) pair. Check that before the insert() array[key] was =null. Ensure that (key, val) is greater than its predecessor and less than its successor.

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Optimized Insert Example Result of the call insert(5, 62)

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Optimized Data Structure Optimized data structure operations: delete(key): Remove the pair pointed to by array[key]. Ensure that array[key]≠null. changekey(key, newval): Call delete(key), followed by insert(key, newval). These calls will check all necessary conditions. deletemin(): Look at the pair that follows the pair (0,sm) (pointed to by array[0]). If no such pair, return “empty”. Else, if there exists pair (key, val), then remove it and set array[key] to null. empty(): Return whether there is a pair following the pair (0,sm).

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Optimized Data Structure Optimized data structure operations: member(key): return whether array[key]=null. predecessor(key): Look at the pair pointed to by array[key]. Follow its backward link to its predecessor pair. If the predecessor pair is (0,sm) then return “smallest”. Else, return the key field of that pair. Note that all the operations can be done in O(1) time.

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Shortest Path A Shortest Path algorithm was implemented using the above algorithm. The original program used the original data structure that produced a certification trail. The checker version was identical to the original except that its data structure was the optimized version that used the trail. Original runtime = O(mlog n) Checker runtime = O(m) (m=number of edges, n=number of nodes)

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Performance: Sort Basic Algorithm – Sorting algorithm with no certification trails. 1 st Execution – Sorter that produces certification trail. 2 nd Execution – Checking algorithm that uses the trail. Speedup – factor of improvement of 2 nd vs Basic. %Savings – of 1 st + 2 nd trails execution over running Basic twice.

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Performance: Sort

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Performance: Convex Hull

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Performance: Shortest Path

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Summary of Experiments The overhead of generating a certification trail is about 2%. The checker run is much faster than the original. It can be run on much slower hardware or use a formally verified language.

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Application to Byzantine Failures Current technique is completely manual. No known way to automatically convert a program to generate a trail. We may develop libraries that use the Certification Trails technique, allowing us to catch errors in a large fraction of a program. Door open to Failure Recovery: when an error is detected the checker goes back to using original code to redo the work.

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