ENCODING NEAREST LARGER VALUES Pat Nicholson* and Rajeev Raman** * MPII ** University of Leicester

Input Data (Relatively Big) DÉJÀ VU: THE ENCODING APPROACH

Input Data (Relatively Big) Preprocess w.r.t. Some Query Encoding (Hope: much smaller)

Input Data (Relatively Big) Preprocess w.r.t. Some Query Encoding (Hope: much smaller) DÉJÀ VU: THE ENCODING APPROACH

Input Data (Relatively Big) Preprocess w.r.t. Some Query Encoding (Hope: much smaller) Auxiliary Data Structures: (Should be smaller still) DÉJÀ VU: THE ENCODING APPROACH

Succinct Data Structure: Minimum Space Possible Encoding (Hope: much smaller) Auxiliary Data Structures: (Should be smaller still) Input Data (Relatively Big) Preprocess w.r.t. Some Query DÉJÀ VU: THE ENCODING APPROACH

Succinct Data Structure: Minimum Space Possible Encoding (Hope: much smaller) Auxiliary Data Structures: (Should be smaller still) Query (Hope: as fast as non- succinct counterpart) Input Data (Relatively Big) Preprocess w.r.t. Some Query DÉJÀ VU: THE ENCODING APPROACH

NEAREST LARGER VALUES

OVERVIEW: ENCODING NLV For all these results: space bound is optimal to within lower order terms DistinctProblemSpaceQNotes Yes Unidirectional Cartesian Tree Bidirectional Cartesian Tree Nondirectional NoUnidirectional [Fischer et al. 2009] Cartesian Tree Bidirectional [Fischer 2011] Schröder Trees (Navigate CSA) Nondirectional

OVERVIEW: ENCODING NLV DistinctProblemSpaceQNotes Yes Unidirectional Cartesian Tree Bidirectional Cartesian Tree NondirectionalThis paper: NLV Tree NoUnidirectional [Fischer et al. 2009] Cartesian Tree Bidirectional [Fischer 2011] Schröder Trees (Navigate CSA) Nondirectional

BIGGER PICTURE

CARTESIAN TREES REVIEW We can rebuild him. We have the technology.

NONDIRECTIONAL NLV TREE Tie breaking rule: break ties to by choosing the one to the right.

TIEBREAKING MATTERS? Rule To the right To the smaller To the larger

IDEA: COMPRESS RUNS

DIGRESSION: PATH (OR CHAIN) COMPRESSION Degree two Degree one Terminal Subtree

COMPRESSING CARTESIAN TREES W.R.T. NLVS Forget about whether it zigs or zags, just store # in prefix…

THE ENCODING

SUB-OPTIMALITY EXAMPLES

LOWER BOUND SKETCH

CONCLUSIONS AND OPEN PROBLEMS

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