Enumeration on Trees under Relabelings
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(2) Problem statement.
(3) Problem: Query evaluation on trees. • Tree on a fixed alphabet • Boolean query to test a property of the tree. 2/19.
(4) Problem: Query evaluation on trees. • Tree on a fixed alphabet • Boolean query to test a property of the tree Example tree <body> <div>. <section> <p>. <h2> <img>. <img>. 2/19.
(5) Problem: Query evaluation on trees. • Tree on a fixed alphabet • Boolean query to test a property of the tree Example query. Example tree <body> <div>. Is there an h2 header and an image that are in the same section?. <section> <p>. <h2> <img>. <img>. 2/19.
(6) Problem: Query evaluation on trees. • Tree on a fixed alphabet • Boolean query to test a property of the tree Example query. Example tree <body> <div>. Is there an h2 header and an image that are in the same section?. <section>. Example answer. <p>. <h2>. →YES <img>. <img>. 2/19.
(7) Problem: Query evaluation on trees. • Tree on a fixed alphabet • Boolean query to test a property of the tree Example query. Example tree <body> <div>. Is there an h2 header and an image that are in the same section?. <section>. Example answer. <p>. <h2>. →YES <img>. <img>. → Theorem: For monadic second-order (MSO) queries, we can check if the query is true or not in linear time in the input tree. 2/19.
(8) Problem: Non-Boolean query evaluation on trees. • Tree on a fixed alphabet • Non-Boolean query to find tuples of nodes satisfying a property. 2/19.
(9) Problem: Non-Boolean query evaluation on trees. • Tree on a fixed alphabet • Non-Boolean query to find tuples of nodes satisfying a property Example tree <body>1 <div>2. <section>3 <h2>4. <p>5 <img>6. <img>7. 2/19.
(10) Problem: Non-Boolean query evaluation on trees. • Tree on a fixed alphabet • Non-Boolean query to find tuples of nodes satisfying a property Example query. Example tree <body>1 <div>2. Find all pairs of an h2 header and an image in the same section. <section>3 <h2>4. <p>5 <img>6. <img>7. 2/19.
(11) Problem: Non-Boolean query evaluation on trees. • Tree on a fixed alphabet • Non-Boolean query to find tuples of nodes satisfying a property Example query. Example tree <body>1 <div>2. Find all pairs of an h2 header and an image in the same section. <section>3 <h2>4. <p>5 <img>6. Example answer → h4, 6i, h4, 7i. <img>7. 2/19.
(12) Problem: Non-Boolean query evaluation on trees. • Tree on a fixed alphabet • Non-Boolean query to find tuples of nodes satisfying a property Example query. Example tree <body>1 <div>2. Find all pairs of an h2 header and an image in the same section. <section>3 <h2>4. <p>5 <img>6. Example answer → h4, 6i, h4, 7i. <img>7. → Corollary: For each possible tuple, we can check in linear time if it is an answer to the query. 2/19.
(13) Enumerating all answers → There can be lots of answers! • “Find all pairs of ...”: output size can be O(|T|2 ). 3/19.
(14) Enumerating all answers → There can be lots of answers! • “Find all pairs of ...”: output size can be O(|T|2 ). 3/19.
(15) Enumerating all answers → There can be lots of answers! • “Find all pairs of ...”: output size can be O(|T|2 ). 3/19.
(16) Enumerating all answers → There can be lots of answers! • “Find all pairs of ...”: output size can be O(|T|2 ). 3/19.
(17) Enumerating all answers → There can be lots of answers! • “Find all pairs of ...”: output size can be O(|T|2 ). 3/19.
(18) Enumerating all answers → There can be lots of answers! • “Find all pairs of ...”: output size can be O(|T|2 ). → Solution: Enumerate answers one after the other 3/19.
(19) Enumeration. Tree ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. 4/19.
(20) Enumeration. Phase 1: Preprocessing Tree ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. 4/19.
(21) Enumeration. Phase 1: Preprocessing Tree. Indexed tree. ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. 4/19.
(22) Enumeration. Phase 1: Preprocessing Tree ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. Indexed tree. Phase 2: Enumeration. 4/19.
(23) Enumeration. Phase 1: Preprocessing Indexed tree. Tree ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. Phase 2: Enumeration. . h4, 6i, Results 4/19.
(24) Enumeration. Phase 1: Preprocessing Indexed tree. Tree ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. 0011 State. Phase 2: Enumeration. . h4, 6i, Results 4/19.
(25) Enumeration. Phase 1: Preprocessing Indexed tree. Tree ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. 0011 State. Phase 2: Enumeration. . h4, 6i, Results 4/19.
(26) Enumeration. Phase 1: Preprocessing Indexed tree. Tree ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. ⊥ State. Phase 2: Enumeration. . h4, 6i, h4, 7i Results 4/19.
(27) Enumeration. Phase 1: Preprocessing Indexed tree. Tree ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y) Query. ⊥ State. Theorem (Bagan’06; Kazana & Segoufin’13) We can enumerate the answers of any MSO query with preprocessing linear in the input tree and constant delay between each answer. Phase 2: Enumeration. . h4, 6i, h4, 7i Results 4/19.
(28) Handling updates. Phase 1: Preprocessing Tree T. Indexed tree. • Trees are often updated, even after we have preprocessed them. 5/19.
(29) Handling updates. Phase 1: Preprocessing Tree T. Indexed tree. • Trees are often updated, even after we have preprocessed them. 5/19.
(30) Handling updates. Phase 1: Preprocessing Tree T. Indexed tree. • Trees are often updated, even after we have preprocessed them. 5/19.
(31) Handling updates. Phase 1: Preprocessing Indexed tree. Tree T. • Trees are often updated, even after we have preprocessed them Work. Data. Delay. Updates. [Bagan, 2006],. trees. O(1). O(T) (from scratch). [Kazana and Segoufin, 2013]. 5/19.
(32) Handling updates. Phase 1: Preprocessing Indexed tree. Tree T. • Trees are often updated, even after we have preprocessed them Work. Data. Delay. Updates. [Bagan, 2006],. trees. O(1). O(T) (from scratch). trees. O(log2 T) O(log2 T). [Kazana and Segoufin, 2013] [Losemann and Martens, 2014]. 5/19.
(33) Handling updates. Phase 1: Preprocessing Indexed tree. Tree T. • Trees are often updated, even after we have preprocessed them Work. Data. Delay. Updates. [Bagan, 2006],. trees. O(1). O(T) (from scratch). [Kazana and Segoufin, 2013] [Losemann and Martens, 2014] [Losemann and Martens, 2014]. trees O(log2 T) O(log2 T) words O(log T) O(log T) 5/19.
(34) Handling updates. Phase 1: Preprocessing Indexed tree. Tree T. • Trees are often updated, even after we have preprocessed them Work. Data. Delay. Updates. [Bagan, 2006],. trees. O(1). O(T) (from scratch). [Kazana and Segoufin, 2013] [Losemann and Martens, 2014] [Losemann and Martens, 2014] [Niewerth and Segoufin, 2018]. trees O(log2 T) O(log2 T) words O(log T) O(log T) words O(1) O(log T) 5/19.
(35) Relabelings. • We focus on relabeling updates: change the label of a node. 6/19.
(36) Relabelings. • We focus on relabeling updates: change the label of a node. • Example: relabel node 7 to <video>. 6/19.
(37) Relabelings. • We focus on relabeling updates: change the label of a node. • Example: relabel node 7 to <video>. 6/19.
(38) Relabelings. • We focus on relabeling updates: change the label of a node. • Example: relabel node 7 to <video> • The tree’s structure never changes. 6/19.
(39) What are relabelings good for?. • Parameterized queries:. • Example: “Find all images in a user-selected section”. <body>1 <div>2. <section>3 <h2>4. <p>5 <img>6 <img>7. 7/19.
(40) What are relabelings good for?. • Parameterized queries:. • Example: “Find all images in a user-selected section” → Write down the user parameters as labels on the tree. <body>1 <div>2. <section>3 <h2>4. <p>5 <img>6 <img>7. 7/19.
(41) What are relabelings good for?. • Parameterized queries:. • Example: “Find all images in a user-selected section” → Write down the user parameters as labels on the tree → Relabel when they change. <body>1 <div>2. <section>3 <h2>4. <p>5 <img>6 <img>7. 7/19.
(42) What are relabelings good for?. • Parameterized queries:. • Example: “Find all images in a user-selected section” → Write down the user parameters as labels on the tree → Relabel when they change. • Group-by with aggregation:. • Example: “For each section, what is the total size of images”. <body>1 <div>2. <section>3 <h2>4. <p>5 <img>6 <img>7. 7/19.
(43) What are relabelings good for?. • Parameterized queries:. • Example: “Find all images in a user-selected section” → Write down the user parameters as labels on the tree → Relabel when they change. • Group-by with aggregation:. • Example: “For each section, what is the total size of images” → Enumerate the groups and write down each group. <body>1 <div>2. <section>3 <h2>4. <p>5 <img>6 <img>7. 7/19.
(44) What are relabelings good for?. • Parameterized queries:. • Example: “Find all images in a user-selected section” → Write down the user parameters as labels on the tree → Relabel when they change. • Group-by with aggregation:. • Example: “For each section, what is the total size of images” → Enumerate the groups and write down each group → Relabel when switching groups. <body>1 <div>2. <section>3 <h2>4. <p>5 <img>6 <img>7. 7/19.
(45) Main result Theorem (Main result) We can enumerate the answers of any MSO query with preprocessing linear in the input tree T and constant delay between each answer and we can relabel any node and update the index in time O(log T). 8/19.
(46) Main result Theorem (Main result) We can enumerate the answers of any MSO query with preprocessing linear in the input tree T and constant delay between each answer and we can relabel any node and update the index in time O(log T) Work. Data. Delay. Updates. [Bagan, 2006],. trees. O(1). N/A. [Kazana and Segoufin, 2013] [Losemann and Martens, 2014] [Losemann and Martens, 2014] [Niewerth and Segoufin, 2018]. trees O(log2 T) O(log2 T) words O(log T) O(log T) words O(1) O(log T). 8/19.
(47) Main result Theorem (Main result) We can enumerate the answers of any MSO query with preprocessing linear in the input tree T and constant delay between each answer and we can relabel any node and update the index in time O(log T) Work. Data. Delay. Updates. [Bagan, 2006],. trees. O(1). N/A. trees words words trees. O(log2 T) O(log T) O(1) O(1). O(log2 T) O(log T) O(log T) O(log T) (relabelings). [Kazana and Segoufin, 2013] [Losemann and Martens, 2014] [Losemann and Martens, 2014] [Niewerth and Segoufin, 2018] [This work]. 8/19.
(48) Main result Theorem (Main result) We can enumerate the answers of any MSO query with preprocessing linear in the input tree T and constant delay between each answer and we can relabel any node and update the index in time O(log T) Work. Data. Delay. Updates. [Bagan, 2006],. trees. O(1). N/A. trees words words trees. O(log2 T) O(log T) O(1) O(1). O(log2 T) O(log T) O(log T) O(log T) (relabelings). [Kazana and Segoufin, 2013] [Losemann and Martens, 2014] [Losemann and Martens, 2014] [Niewerth and Segoufin, 2018] [This work]. → Consequences for group-by, aggregation, parameterized queries. 8/19.
(49) Proof techniques.
(50) Knowledge compilation To make the proof modular, we follow knowledge compilation:. Tree. Phase 1: Preprocessing Indexed tree. Phase 2: Enumeration. hx : 4, y : 6i, hx : 4, y : 7i. Results. ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y). Query. 9/19.
(51) Knowledge compilation To make the proof modular, we follow knowledge compilation:. • Preprocessing: Compute a circuit representation of the answers • Enumeration: Apply a generic algorithm on the circuit ×. Phase 1: Preprocessing Tree. x:4. ∪ y:6. y:7. Circuit. Phase 2: Enumeration. hx : 4, y : 6i, hx : 4, y : 7i. Results. ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y). Query. 9/19.
(52) Knowledge compilation To make the proof modular, we follow knowledge compilation:. • Preprocessing: Compute a circuit representation of the answers • Enumeration: Apply a generic algorithm on the circuit ×. Phase 1: Preprocessing Tree. x:4. ∪ y:6. y:7. Circuit. Phase 2: Enumeration. hx : 4, y : 6i, hx : 4, y : 7i. Results. ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y). Query First present approach without relabelings (as in our ICALP’17 paper) then extend the approach to support relabelings 9/19.
(53) Set circuits A set circuit represents a set of answers to a query Q(x, y). 10/19.
(54) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6”. 10/19.
(55) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons. 10/19.
(56) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i. 10/19.
(57) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i Three kinds of set-valued gates: ×. x:4. ∪. y:6. y:7 10/19.
(58) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i Three kinds of set-valued gates:. • Variable gate. ×. x:4. :. → captures hx : 4i. x:4. ∪. y:6. y:7 10/19.
(59) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i Three kinds of set-valued gates:. • Variable gate. ×. x:4. :. → captures hx : 4i. x:4. • Union gate. ∪. ∪. :. → union of sets of tuples. y:6. y:7 10/19.
(60) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i Three kinds of set-valued gates:. • Variable gate. ×. :. x:4. → captures hx : 4i. x:4. • Union gate. ∪. :. ∪. → union of sets of tuples. y:6. y:7. • Product gate. ×. :. → relational product 10/19.
(61) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i Three kinds of set-valued gates:. • Variable gate. ×. :. x:4. → captures hx : 4i. x:4. • Union gate. ∪. → union of sets of tuples. hy : 6i y:6. :. ∪. y:7. • Product gate. ×. :. → relational product 10/19.
(62) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i Three kinds of set-valued gates:. • Variable gate. ×. :. x:4. → captures hx : 4i. x:4. • Union gate. ∪. hy : 6i y:6. . hy : 7i. y:7. :. ∪. → union of sets of tuples. • Product gate. ×. :. → relational product 10/19.
(63) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i Three kinds of set-valued gates: × x:4. hy : 6i, hy : 7i. y:6. . hy : 7i. y:7. :. x:4. → captures hx : 4i. • Union gate. ∪. hy : 6i. • Variable gate. :. ∪. → union of sets of tuples. • Product gate. ×. :. → relational product 10/19.
(64) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i Three kinds of set-valued gates: × . . hx : 4i x:4. hy : 6i, hy : 7i. y:6. . hy : 7i. y:7. :. x:4. → captures hx : 4i. • Union gate. ∪. hy : 6i. • Variable gate. :. ∪. → union of sets of tuples. • Product gate. ×. :. → relational product 10/19.
(65) Set circuits A set circuit represents a set of answers to a query Q(x, y). • Singleton x : 6 → “the free variable x is mapped to node 6” • Tuple hx : 4, y : 6i: tuple of singletons • The circuit captures a set of tuples, e.g., hx : 4, y : 6i, hx : 4, y : 7i × . . hx : 4i x:4. Three kinds of set-valued gates:. hx : 4, y : 6i hx : 4, y : 7i hy : 6i, hy : 7i. y:6. . hy : 7i. y:7. :. x:4. → captures hx : 4i. • Union gate. ∪. hy : 6i. • Variable gate. :. ∪. → union of sets of tuples. • Product gate. ×. :. → relational product 10/19.
(66) Preprocessing: Set circuit construction ×. Phase 1: Preprocessing Tree. x:4. ∪ y:6. y:7. Circuit. Phase 2: Enumeration. . hx : 4, y : 6i, hx : 4, y : 7i. Results. ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y). Query Theorem For any MSO query Q(x1 , . . . , xk ), given a tree T, we can build in O(T) a set circuit capturing exactly the set of answers of Q on T: {hx1 : n1 , . . . , xk : nk i | (n1 , . . . , nk ) ∈ T k }. 11/19.
(67) Preprocessing: Set circuit construction ×. Phase 1: Preprocessing Tree. x:4. ∪ y:6. y:7. Circuit. Phase 2: Enumeration. . hx : 4, y : 6i, hx : 4, y : 7i. Results. ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y). Query Theorem For any MSO query Q(x1 , . . . , xk ), given a tree T, we can build in O(T) a set circuit capturing exactly the set of answers of Q on T: {hx1 : n1 , . . . , xk : nk i | (n1 , . . . , nk ) ∈ T k }. • Proof idea: Translate query to bottom-up tree automaton. and build a provenance circuit following the structure of the tree 11/19.
(68) Enumeration on set circuits ×. Phase 1: Preprocessing Tree. x:4. ∪ y:6. y:7. Circuit. Phase 2: Enumeration. . hx : 4, y : 6i, hx : 4, y : 7i. Results. ∃s section(s)∧ s x∧s y∧ h2(x) ∧ img(y). Query. Theorem Given a set circuit satisfying some conditions, we can enumerate all tuples that it captures with linear preprocessing and constant delay E.g., for hx : 4, y : 6i, hx : 4, y : 7i : enumerate hx : 4, y : 6i, then hx : 4, y : 7i 12/19.
(69) General enumeration approach → Enumerate the set T(g) captured by each gate g → Do it by top-down induction on the circuit. 13/19.
(70) General enumeration approach → Enumerate the set T(g) captured by each gate g → Do it by top-down induction on the circuit Base case: variable. x:n. :. 13/19.
(71) General enumeration approach → Enumerate the set T(g) captured by each gate g → Do it by top-down induction on the circuit Base case: variable. x:n. : enumerate hx : ni and stop. 13/19.
(72) General enumeration approach → Enumerate the set T(g) captured by each gate g → Do it by top-down induction on the circuit Base case: variable. x:n. : enumerate hx : ni and stop. ∪-gate ∪ g1. g2. Concatenation: enumerate T(g1 ) and then enumerate T(g2 ). 13/19.
(73) General enumeration approach → Enumerate the set T(g) captured by each gate g → Do it by top-down induction on the circuit Base case: variable. g1. x:n. : enumerate hx : ni and stop. ∪-gate. ×-gate. ∪. × g2. Concatenation: enumerate T(g1 ) and then enumerate T(g2 ). g1. g2. Lexicographic product: for every t1 in T(g1 ): for every t2 in T(g2 ): output t1 + t2 13/19.
(74) Circuit conditions Enumeration relies on some conditions on the input circuit (d-DNNF):. ×. x:4. ∪. y:6. y:7. 14/19.
(75) Circuit conditions Enumeration relies on some conditions on the input circuit (d-DNNF):. •. ∪. are all deterministic:. For any two inputs g1 and g2 of a ∪-gate, the captured sets T(g1 ) and T(g2 ) are disjoint (they have no tuple in common) → Avoids duplicate tuples. ×. x:4. ∪. y:6. y:7. 14/19.
(76) Circuit conditions Enumeration relies on some conditions on the input circuit (d-DNNF):. •. ∪. are all deterministic:. For any two inputs g1 and g2 of a ∪-gate, the captured sets T(g1 ) and T(g2 ) are disjoint (they have no tuple in common) → Avoids duplicate tuples. •. ×. ×. x:4. ∪. are all decomposable:. For any two inputs g1 and g2 of a ×-gate, no variable has a path to both g1 and g2. y:6. y:7. → Avoids duplicate singletons 14/19.
(77) Circuit conditions Enumeration relies on some conditions on the input circuit (d-DNNF):. •. ∪. are all deterministic:. For any two inputs g1 and g2 of a ∪-gate, the captured sets T(g1 ) and T(g2 ) are disjoint (they have no tuple in common) → Avoids duplicate tuples. •. ×. ×. x:4. ∪. are all decomposable:. For any two inputs g1 and g2 of a ×-gate, no variable has a path to both g1 and g2. y:6. y:7. → Avoids duplicate singletons. • Also an additional upwards-determinism condition. 14/19.
(78) Enumeration subtleties. ∪ y:6. × x:4. ∪. • We must not waste time in gates capturing ∅. 15/19.
(79) Enumeration subtleties. ∪ y:6. × x:4. ∪. • We must not waste time in gates capturing ∅ → Label them during the preprocessing. 15/19.
(80) Enumeration subtleties × ×. ∪ y:6. × x:4. ∪. × ×. × ×. x:4. • We must not waste time in gates capturing ∅ → Label them during the preprocessing. • We must not waste time because of gates capturing. . hi. 15/19.
(81) Enumeration subtleties × ×. ∪ y:6. × x:4. ∪. × ×. × ×. x:4. • We must not waste time in gates capturing ∅ → Label them during the preprocessing. • We must not waste time because of gates capturing. . hi. → Homogenization to set them aside. 15/19.
(82) Enumeration subtleties × ×. ∪ y:6. × x:4. ∪. ∪ ×. ×. ∪ ×. ×. ··· x:4. ∪ ∪. ···. ∪ ··· ···. • We must not waste time in gates capturing ∅ → Label them during the preprocessing. • We must not waste time because of gates capturing. . hi. → Homogenization to set them aside. • We must not waste time in hierarchies of ∪-gates 15/19.
(83) Enumeration subtleties × ×. ∪ y:6. × x:4. ∪. ∪ ×. ×. ∪ ×. ×. ··· x:4. ∪ ∪. ···. ∪ ··· ···. • We must not waste time in gates capturing ∅ → Label them during the preprocessing. • We must not waste time because of gates capturing. . hi. → Homogenization to set them aside. • We must not waste time in hierarchies of ∪-gates. → Precompute a reachability index (uses upwards-determinism) 15/19.
(84) Hybrid circuits to support relabelings To support relabelings, we use hybrid circuits that have:. • •. Boolean gates that depend only on the labeling Set gates that capture a set of tuples for each labeling. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 16/19.
(85) Hybrid circuits to support relabelings To support relabelings, we use hybrid circuits that have:. • •. Boolean gates that depend only on the labeling Set gates that capture a set of tuples for each labeling Four kinds of Boolean gates: ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 16/19.
(86) Hybrid circuits to support relabelings To support relabelings, we use hybrid circuits that have:. • •. Boolean gates that depend only on the labeling Set gates that capture a set of tuples for each labeling Four kinds of Boolean gates:. • Variable gate. . → true iff node 4 is labeled h2. ×. h2 : 4. hx : 4i. ∪ img : 6. h2 : 4 :. hy : 6i. img : 7. hy : 7i 16/19.
(87) Hybrid circuits to support relabelings To support relabelings, we use hybrid circuits that have:. • •. Boolean gates that depend only on the labeling Set gates that capture a set of tuples for each labeling Four kinds of Boolean gates: . h2 : 4 :. • AND, OR, NOT. ∧. → true iff node 4 is labeled h2. ×. h2 : 4. ∨. ¬. :. → usual semantics. hx : 4i. ∪ img : 6. • Variable gate. hy : 6i. img : 7. hy : 7i 16/19.
(88) Hybrid circuits to support relabelings To support relabelings, we use hybrid circuits that have:. • •. Boolean gates that depend only on the labeling Set gates that capture a set of tuples for each labeling Four kinds of Boolean gates: ×. • AND, OR, NOT. ∧. ∨. :. ¬. → usual semantics. hx : 4i One new set-valued gate:. ∪ . •. hy : 6i. h2 : 4 :. → true iff node 4 is labeled h2. h2 : 4. img : 6. • Variable gate. img : 7. hy : 7i. . : Test gate. → ∅ if Boolean input is false → like the set input otherwise 16/19.
(89) Hybrid circuit semantics. • For every labeling, the hybrid circuit captures a set of tuples. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 17/19.
(90) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 17/19.
(91) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i hy : 6i. img : 7. hy : 7i 17/19.
(92) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i hy : 6i. img : 7. hy : 7i hy : 7i 17/19.
(93) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i. ×. h2 : 4 . hy : 6i, hy : 7i . img : 6. hx : 4i. ∪. hy : 6i hy : 6i. img : 7. hy : 7i hy : 7i 17/19.
(94) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i. . ×. h2 : 4 . hy : 6i, hy : 7i . img : 6. hx : 4, y : 6i, hx : 4, y : 7i. hx : 4i. ∪. hy : 6i hy : 6i. img : 7. hy : 7i hy : 7i 17/19.
(95) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i • When the tree is relabeled, change the Boolean variables. . ×. h2 : 4 . hy : 6i, hy : 7i . img : 6. hx : 4, y : 6i, hx : 4, y : 7i. hx : 4i. ∪. hy : 6i hy : 6i. img : 7. hy : 7i hy : 7i 17/19.
(96) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i • When the tree is relabeled, change the Boolean variables. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 17/19.
(97) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i • When the tree is relabeled, change the Boolean variables. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i hy : 6i. img : 7. hy : 7i 17/19.
(98) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i • When the tree is relabeled, change the Boolean variables. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i hy : 6i. img : 7. ∅ hy : 7i 17/19.
(99) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i • When the tree is relabeled, change the Boolean variables. ×. h2 : 4 . hy : 6i . img : 6. hx : 4i. ∪ hy : 6i. hy : 6i. img : 7. ∅ hy : 7i 17/19.
(100) Hybrid circuit semantics labeling, the hybrid circuit captures a set of tuples • For→ every Here, the captured set is hx : 4, y : 6i, hx : 4, y : 7i the tree is relabeled, change the Boolean variables • When → New captured set: hx : 4, y : 6i. . . ×. h2 : 4 . hy : 6i . img : 6. hx : 4, y : 6i. hx : 4i. ∪ hy : 6i. hy : 6i. img : 7. ∅ hy : 7i 17/19.
(101) Complexity of relabelings. • When a label changes, update the circuit bottom-up. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 18/19.
(102) Complexity of relabelings. • When a label changes, update the circuit bottom-up. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 18/19.
(103) Complexity of relabelings. • When a label changes, update the circuit bottom-up. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 18/19.
(104) Complexity of relabelings. • When a label changes, update the circuit bottom-up • The circuit follows the structure of the input tree T so updates are in O(height(T)). ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 18/19.
(105) Complexity of relabelings. • When a label changes, update the circuit bottom-up • The circuit follows the structure of the input tree T. so updates are in O(height(T)) → Balancing lemma: Rewrite the input tree to make it balanced. ×. h2 : 4. hx : 4i. ∪ img : 6. hy : 6i. img : 7. hy : 7i 18/19.
(106) Conclusion.
(107) Summary and future work Summary:. • Problem: enumerate the answers of an MSO query on a tree with efficient support for relabeling updates on the tree. 19/19.
(108) Summary and future work Summary:. • Problem: enumerate the answers of an MSO query on a tree with efficient support for relabeling updates on the tree. • Main result: we can do this with linear preprocessing,. constant delay between each answer, and log update time. 19/19.
(109) Summary and future work Summary:. • Problem: enumerate the answers of an MSO query on a tree with efficient support for relabeling updates on the tree. • Main result: we can do this with linear preprocessing,. constant delay between each answer, and log update time. • Consequences: group-by, parameterized queries, aggregation. 19/19.
(110) Summary and future work Summary:. • Problem: enumerate the answers of an MSO query on a tree with efficient support for relabeling updates on the tree. • Main result: we can do this with linear preprocessing,. constant delay between each answer, and log update time. • Consequences: group-by, parameterized queries, aggregation Future work:. • Practice: implement the technique with automata • Applications: text extraction? e.g., document spanners (ongoing) • Updates: support insertions/deletions? (ongoing) 19/19.
(111) Summary and future work Summary:. • Problem: enumerate the answers of an MSO query on a tree with efficient support for relabeling updates on the tree. • Main result: we can do this with linear preprocessing,. constant delay between each answer, and log update time. • Consequences: group-by, parameterized queries, aggregation Future work:. • Practice: implement the technique with automata • Applications: text extraction? e.g., document spanners (ongoing) • Updates: support insertions/deletions? (ongoing) Thanks for your attention! 19/19.
(112) References i. Bagan, G. (2006). MSO queries on tree decomposable structures are computable with linear delay. In CSL. Kazana, W. and Segoufin, L. (2013). Enumeration of monadic second-order queries on trees. TOCL, 14(4). Losemann, K. and Martens, W. (2014). MSO queries on trees: enumerating answers under updates. In CSL-LICS..
(113) References ii. Niewerth, M. and Segoufin, L. (2018). Enumeration of MSO queries on strings with constant delay and logarithmic updates. In PODS. To appear..
(114) Set circuit construction. • Automaton: “Select all node pairs (x, y)”. • States: {∅, x, y, xy}.
(115) Set circuit construction. • Automaton: “Select all node pairs (x, y)” n. • States: {∅, x, y, xy}.
(116) Set circuit construction. • Automaton: “Select all node pairs (x, y)”. • States: {∅, x, y, xy}. n. x:n.
(117) Set circuit construction. • Automaton: “Select all node pairs (x, y)” n. • States: {∅, x, y, xy}. ∅. x. y. xy. ∪. ∪. ∪. ∪. x:n ∅. x. y. xy. ∅. x. y. xy. ∪. ∪. ∪. ∪. ∪. ∪. ∪. ∪.
(118) Set circuit construction. • Automaton: “Select all node pairs (x, y)” n. • States: {∅, x, y, xy}. ∅. x. y. xy. ∪. ∪. ∪. ∪. ×. x:n. ∅. x. y. xy. ∅. x. y. xy. ∪. ∪. ∪. ∪. ∪. ∪. ∪. ∪.
(119) Set circuit construction. • Automaton: “Select all node pairs (x, y)” n. • States: {∅, x, y, xy}. ∅. x. y. xy. ∪. ∪. ∪. ∪. ×. x:n. ∅. x. y. xy. ∅. x. y. xy. ∪. ∪. ∪. ∪. ∪. ∪. ∪. ∪.
(120) Set circuit construction. • Automaton: “Select all node pairs (x, y)” n. • States: {∅, x, y, xy}. ∅. x. y. xy. ∪. ∪. ∪. ∪ ×. ×. x:n. ∅. x. y. xy. ∅. x. y. xy. ∪. ∪. ∪. ∪. ∪. ∪. ∪. ∪.
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