High Average-Utility Sequential Mining

• HAUSPM is defined as a sequential pattern mining approach that uses average utility, taking sequence length into account to mitigate bias.
• It normalizes utility to yield shorter, more typical patterns that align with user-specified queries for improved relevance.
• Algorithms like HAUSP-PG and TAUSQ-PG implement dual pruning and projection-based scanning for efficient and targeted mining.

High Average-Utility Sequential Pattern Mining (HAUSPM) is a utility-oriented formulation of sequential pattern mining in which the evaluation criterion is average utility rather than total utility. In the recent literature, HAUSPM is presented as a way to mitigate the length bias of High-Utility Sequential Pattern Mining (HUSPM), since longer sequences can accumulate large total utility merely by containing more items. By normalizing utility with respect to sequence length, HAUSPM is described as fairer for longer patterns and more conducive to interpretability. The 2025 literature further extends this setting to targeted querying, where only patterns containing a user-specified sequence are retained, leading to targeted high average-utility sequential pattern mining and algorithms such as HAUSP-PG and TAUSQ-PG (Cao et al., 5 Oct 2025, Cao et al., 11 Oct 2025).

1. Conceptual scope and motivation

Traditional frequent sequential pattern mining identifies patterns that occur often in a sequence database. HUSPM augments this setting by incorporating utility, such as profit, importance, or risk, and thereby overcomes frequency bias. The drawback emphasized in the HAUSPM literature is that HUSPM tends to favor long and complicated patterns, because longer sequences accumulate more utility and are therefore advantaged under absolute utility criteria. HAUSPM addresses this limitation by using average utility, which directly takes pattern length into account (Cao et al., 5 Oct 2025).

This motivation is especially prominent for long-sequence settings. One 2025 study explicitly notes that, in contrast to retail business analysis, applications such as cybersecurity or artificial intelligence often involve much longer sequences, and that pruning strategies therefore have a more pronounced impact on efficiency. Another study makes a closely related point from the targeted-mining perspective: utility-based methods often generate a large number of long and complicated sequences, whereas average utility and target constraints together improve pattern relevance, usability, and interactivity by retaining only patterns aligned with user-specified goals (Cao et al., 5 Oct 2025, Cao et al., 11 Oct 2025).

The interpretive consequence is central. In the targeted formulation, average utility is described as producing patterns that are shorter, more typical, and more relevant to the specified signal of interest. This suggests that HAUSPM is not only a search-space reduction device but also an attempt to redefine what counts as an informative sequential pattern under utility-oriented semantics (Cao et al., 11 Oct 2025).

2. Formal definitions and problem statements

The core formalism is based on quantitative sequence data. Let $I = \{i_1, ..., i_M\}$ denote the set of items, and let a quantitative sequence be written as $QS = \langle Y_1, ..., Y_n \rangle$, where each $Y_j$ is an itemset containing quantitative items $(i, q)$. For a quantitative item, the utility is defined as

$u(i,j,QS) = q(i,j,QS) \times eu(i),$

where $eu(i)$ is the external utility. The utility of an itemset $Y_j$ is

$$u(Y_j, j, QS) = \sum_{(i, q) \in Y_j} q(i, j, QS) \times eu(i),$$

and the utility of the q-sequence is

$u(QS) = \sum_{Y_j \in QS} u(Y_j, j, QS).$

The average utility of a q-sequence is then

$au(QS) = \frac{u(QS)}{|QS|},$

where $QS = \langle Y_1, ..., Y_n \rangle$0 is the total number of items in the sequence (Cao et al., 11 Oct 2025).

For a sequential pattern $QS = \langle Y_1, ..., Y_n \rangle$1 that appears in $QS = \langle Y_1, ..., Y_n \rangle$2, the literature defines instance utility $QS = \langle Y_1, ..., Y_n \rangle$3 for an occurrence position $QS = \langle Y_1, ..., Y_n \rangle$4, and instance average utility

$QS = \langle Y_1, ..., Y_n \rangle$5

Over all instances of $QS = \langle Y_1, ..., Y_n \rangle$6 in $QS = \langle Y_1, ..., Y_n \rangle$7, the sequence-level average utility is

$QS = \langle Y_1, ..., Y_n \rangle$8

The HAUSPM literature also states the database-wide criterion in thresholded form: a pattern is a high average-utility sequential pattern if its database-level average utility reaches the minimum average utility threshold scaled by total database utility, $QS = \langle Y_1, ..., Y_n \rangle$9 (Cao et al., 5 Oct 2025).

The targeted formulation refines this model. Given a q-sequence database $Y_j$0, a query sequence $Y_j$1, and a minimum average utility threshold $Y_j$2, the database is first filtered to

$Y_j$3

The minimum acceptable average utility becomes $Y_j$4. The objective is then to find all patterns $Y_j$5 such that $Y_j$6 and the average utility of $Y_j$7, measured over the filtered database, is at least $Y_j$8 (Cao et al., 11 Oct 2025).

3. Upper bounds, pruning, and pattern-growth search

Because HAUSPM operates over an exponentially large sequential search space, its practical development is inseparable from upper bounds and pruning strategies. The 2025 HAUSP-PG algorithm is built around a dual pruning strategy that independently processes the current pattern prefix and the remaining suffix. Prefix pruning discards a current sequence if it or any extension cannot possibly be high average-utility, while suffix pruning filters unpromising items in the remaining subsequence, keeping only those that can potentially raise the average utility above threshold (Cao et al., 5 Oct 2025).

A central contribution of HAUSP-PG is sorting-free upper-bound estimation. Earlier approaches are characterized as relying either on maximum-item-utility bounds, which can grossly overestimate when most items are low-utility, or on top-$Y_j$9 item utilities that require repeated sorting as patterns grow. HAUSP-PG replaces such repeated sorting with projection-based scanning. Its Prefix Extension Average Utility upper bound is given as

$(i, q)$0

and related bounds, including RSAU and TRSAU, are calculated incrementally by scanning the suffix rather than sorting it. The stated complexity consequence is that the suffix can be processed in $(i, q)$1 time rather than $(i, q)$2, where $(i, q)$3 is suffix length (Cao et al., 5 Oct 2025).

The targeted HAUSPM literature develops a complementary family of bounds. TAUSQ-PG uses S-Extension and I-Extension pruning, depth pruning, and width pruning. For depth pruning it defines Suffix Remaining Average Utility: $(i, q)$4 If $(i, q)$5, then no extension of $(i, q)$6 can yield a valid targeted high average-utility sequential pattern. Width pruning is based on TDAU, the terminated descendants average utility; further variants such as vSRAU and vTDAU refine these bounds by focusing on relevant parts of the query suffix and reducing data-structure overhead (Cao et al., 11 Oct 2025).

The common pattern across these systems is that HAUSPM does not rely on raw enumeration. It relies on upper bounds that are sufficiently tight to control candidate growth while remaining cheap enough to evaluate repeatedly during recursive pattern growth.

4. Targeted HAUSPM and the TAUSQ-PG framework

Targeted HAUSPM adds a user-centric constraint to the mining process: only patterns containing a specified query sequence $(i, q)$7 are of interest. This alters both preprocessing and recursion. At preprocessing time, all sequences not containing $(i, q)$8 are removed. During mining, candidate patterns are pruned whenever they cannot possibly contain the remaining query suffix. The literature presents this as an efficiency mechanism and also as a usability mechanism, because it retains only patterns containing user-specified goals (Cao et al., 11 Oct 2025).

TAUSQ-PG operationalizes this formulation through explicit query-tracking devices. Two flags, IMatch and IIMatch, track the matching progress of a candidate pattern with the query sequence using position-based status. The notions of query suffix ($(i, q)$9) and query prefix ($u(i,j,QS) = q(i,j,QS) \times eu(i),$0) constrain recursion: at every step, the algorithm considers only those growth extensions that can still match the remaining query suffix. Prefix pattern pruning follows directly from this logic; if the remaining subsequence cannot possibly contain $u(i,j,QS) = q(i,j,QS) \times eu(i),$1, the branch is pruned (Cao et al., 11 Oct 2025).

This targeted regime introduces a different search objective from untargeted HAUSPM. The task is no longer to enumerate all high average-utility sequential patterns in $u(i,j,QS) = q(i,j,QS) \times eu(i),$2, but to enumerate only those high average-utility patterns in the filtered database $u(i,j,QS) = q(i,j,QS) \times eu(i),$3 that also satisfy $u(i,j,QS) = q(i,j,QS) \times eu(i),$4. A plausible implication is that TAUSQ-PG turns relevance into a first-class constraint rather than treating it as a post hoc filtering step.

The same work also defines specialized evaluation metrics for this setting: runtime, candidate set size, and peak memory usage. These metrics are used specifically to assess whether targeted querying and average-utility normalization jointly reduce redundant sequence generation in large-scale and long-sequence datasets (Cao et al., 11 Oct 2025).

5. Data structures and implementation frameworks

HAUSPM algorithms depend heavily on compact positional and utility-aware representations. TAUSQ-PG uses three specialized structures. The Q-matrix encodes the utility and structure of each sequence. The Targeted Chain is a compact projected representation specific to each prefix pattern and stores pattern-growth status together with upper-bound calculations. The LI-Table supports fast position-based query suffix checking and is used in conjunction with IMatch and IIMatch (Cao et al., 11 Oct 2025).

HAUSP-PG uses a projection-based structure denoted proDB, together with seqArray and exList. Its mining process is organized as pattern growth over a lexicographically ordered sequence tree, the LQS-tree. The key implementation claim is that projected structures store the utility information needed for average-utility evaluation and upper-bound computation without requiring repeated suffix sorting. This design is described as no-redundant and as especially advantageous for long sequences (Cao et al., 5 Oct 2025).

Related utility-mining systems illuminate the broader engineering context. ProUM, a projection-based utility mining method for HUSPM, uses a utility-array to store sequence order, utility, remaining utility, and next-position information, and it applies projection to utility-arrays rather than to raw projected databases (Gan et al., 2019). USPT, which addresses high-utility sequential patterns with individualized thresholds, combines a lexicographic-sequential tree with a utility-array structure and I-Concatenation and S-Concatenation growth operators (Gan et al., 2019). These are not HAUSPM algorithms, but they show that array-based and projection-based sequence representations are a recurring architectural pattern in utility-oriented sequence mining.

6. Empirical behavior and comparative results

The empirical studies reported for HAUSPM emphasize runtime, memory usage, candidate set size, and scalability. TAUSQ-PG is evaluated on four real datasets—Bible, Leviathan, Sign, and Kosarak—and two synthetic datasets, $u(i,j,QS) = q(i,j,QS) \times eu(i),$5 and $u(i,j,QS) = q(i,j,QS) \times eu(i),$6. Its baselines are $u(i,j,QS) = q(i,j,QS) \times eu(i),$7, which performs recursive targeted querying at each search step, and $u(i,j,QS) = q(i,j,QS) \times eu(i),$8, which only preprocesses the database to filter target sequences and performs no recursive filtering during search (Cao et al., 11 Oct 2025).

Across those experiments, TAUSQ-PG is reported to consistently outperform its baselines by significant margins in runtime and memory, especially for large and long-sequence datasets. The improvement is stated to be most dramatic at lower average-utility thresholds, where candidate sets balloon in non-targeted methods. It also generates the smallest candidate set, and this is explicitly connected to interpretability: fewer, more relevant, and compact patterns are returned because of the joint use of average utility and targeted query constraints. When target query length increases, TAUSQ-PG remains manageable in runtime and memory whereas the baselines degrade rapidly (Cao et al., 11 Oct 2025).

HAUSP-PG is evaluated on real-life datasets including Sign, Bible, Leviathan, Yoochoose, and Kosarak, as well as synthetic datasets. Its reported behavior is that it outperforms or matches state-of-the-art approaches, notably SimEHAUSM and HUSP-SP, especially on large, long, or skewed datasets. The number of candidate sequences is substantially reduced by prefix and suffix pruning, memory usage is lower or comparable because compact data structures eliminate large sort buffers, and scalability is described as nearly linear in database size when $u(i,j,QS) = q(i,j,QS) \times eu(i),$9 is fixed. The ablation study further reports that different upper-bound variants behave differently across datasets, but the hybrid upper-bound design is best or close to best universally (Cao et al., 5 Oct 2025).

These findings collectively present HAUSPM as an algorithmic efficiency problem as much as an interestingness-measure problem. The mining criterion alone does not guarantee tractability; tractability arises from the combination of length-aware utility, projection-based data structures, and aggressive pruning.

7. Relation to adjacent utility-oriented sequence mining

HAUSPM belongs to a wider family of utility-oriented sequence mining tasks, but it is not interchangeable with them. HUSPM uses total utility and is explicitly described as biased toward long patterns. HAUSPM addresses that bias by normalizing with respect to pattern length. HUOSPM, by contrast, uses utility occupancy, defining

$eu(i)$0

and then averaging this occupancy over supporting sequences. In that literature, utility occupancy is contrasted with HAUSPM as a relative, within-sequence notion of importance rather than an average absolute utility notion (Huang et al., 2022).

Another neighboring formulation is utility mining with individualized thresholds. USPT assigns each item its own minimum utility threshold, uses $eu(i)$1 and $eu(i)$2, and applies upper-bound pruning in a lexicographic-sequential tree. Its relation to HAUSPM is not metric equivalence but shared methodology: both attempt to break the limitations of a single global minimum-utility criterion by introducing alternative evaluation semantics and new pruning rules (Gan et al., 2019).

Targeted sequential rule mining offers a further comparison point. Guided exploration of sequential rules introduces database reduction, target queries, and tight upper bounds for both frequency-based and utility-based sequential rule mining. This suggests that user-centric relevance constraints, which are central to targeted HAUSPM, are part of a broader shift in sequence mining away from exhaustive result sets and toward concise, query-aligned discovery (Gan et al., 6 Feb 2026).

A notable source of ambiguity in adjacent literature is that not all normalization schemes are identical. The 2025 HAUSPM papers formalize average utility primarily through length normalization, whereas the HUOSPM comparison section characterizes average-utility mining as averaging utility over supporting sequences. This suggests that, in utility-oriented sequence mining, the exact meaning of “average” remains formulation-dependent and should be read from the paper’s own definitions rather than inferred from the label alone (Cao et al., 5 Oct 2025, Huang et al., 2022).